This answer is now complete. Note all values of cumulative CO2 produced refer to as of end of year 2018.
I have finally found an ExxonMobil document that cited a source, rather than simply state the claim (as is with this and this ExxonMobil site). The document is here and the claim is mentioned in the first paragraph of page 20 of the pdf. Part of the paragraph is cited below.
Since 1970, ExxonMobil has cumulatively captured more CO2 than any other company - accounting for more than 40 percent of cumulative CO2 captured. We have a working interest in more than one-fifth of the world’s carbon capture capacity, capturing nearly 7 million tonnes of CO2 in 2017.
The source, listed as number 28 on page 38, reads
Source: Global CCS Institute. Data updated as of April 2018 and based on cumulative anthropogenic carbon dioxide capture volume. Anthropogenic CO2, for the purposes of this calculation, means CO2 that without carbon capture and storage would have been emitted to the atmosphere, including,but not limited to: reservoir CO2 from gas fields; CO2 emitted during production and CO2 emitted during combustion. It does not include natural CO2 produced solely for enhanced oil recovery.
This source isn't specific and I could not find any document from the Global CCS Institute that mentioned "cumulative anthropogenic carbon dioxide capture volume." Note that ExxonMobil is a member of the Global CCS Institute, so there may be a conflict of interest. ("Amongst [the Insititute's] representation are ... multinationals such as ... ExxonMobil...") However, "the governments of the United States, the United Kingdom, China, Japan and Australia" are members, so, for this answer, I will assume this source is credible.
I was unable to find a document from the Global CCS Institute that contained data on "cumulative anthropogenic carbon dioxide capture volume." I did, however, find this document. On page 55, under "Largest CCS integrated project in operation" is the "ExxonMobil Shute Creek CCS-EOR project in North America." Furthermore, a bullet point reads "Capture and store 6.5MtCO2/year." Thus, this may be the "about 7 million tonnes per year of CO2." On page 22, however, it lists "7 MtCO2/year" as the capacity of the plant.
This EPA document writes "ExxonMobil forecasts the total volume of CO2 stored [at the ExxonMobil Shute Creek Treating Facility] over the modeled injection period to be approximately 37 million metric tons." but doesn't state an expected million metric ton per year of injected CO2. Additionally, this MIT site lists the "size" as "7 Mt/yr (365m cubic feet/day)."
In more words, but with a similar meaning, zeroco2.no wrote
ExxonMobil's Shute Creek gas processing plant near LaBarge, Wyoming, is currently capturing around 7 million tonnes per annum of CO2.
From the Global CCS Institute:
CO2 CAPTURE CAPACITY VOLUME: 7 million tonnes per annum (Mtpa)
1986: Exxon Shute Creek Gas Processing Facility in Wyoming. This natural gas processing plant serves ExxonMobil, Chevron, and Anadarko Petroleum CO2 pipeline systems to oil fields in Wyoming and Colorado and is the largest commercial carbon capture facility in the world at 7 million tons of capacity annually.
Thus, the first claim is true if the Exxon Shute Creek Gas Processing Facility captures 7 million metric tons of CO2 annually. I will need to further look into the credibility of the sources, but the quantity and agreement of the sources seems to make falsification of the value (7 million tons) doubtful. It is interesting to note that the facility is operated to benefit ExxonMobil (see enhanced oil recovery) and not necessarily to reduce carbon emissions or be environmentally friendly.
Speculation on 2: page 22 of this document lists "All integrated projects in operation are associated with the oil and gas industry." Assuming that ExxonMobil's Shute Creek plant has been capturing CO2 since 1986 (as seen in the document) at an average rate of 4 MtCO2/year, the plant would have captured 132 MtCO2 now (4*33). By performing similar math for all plants listed on that page, all other plants have captured 160 MtCO2. Thus, ExxonMobil will have captured 45% (which is more than 40%) of all cumulative CO2 captured.
Regarding 2: The Global CCS Institute has an "up to date" database of CCS facilities. They write
Large-scale integrated CCS facilities comprise the capture, transport, and storage of CO2 at a scale of at least 800,000 tonnes of CO2 annually for a coal-based power plant, or at least 400,000 tonnes of CO2 annually for other emissions-intensive industrial facilities (including natural gas-based power generation).
Setting "Category" on the website to "Large-scale CCS facilities" and sorting by "Operation Date," we can count up the total metric tons per year of all captured CO2 in industrial plants. Doing this only for plants "in operation," 33.3 million metric tons of CO2 is captured annually (please check my work!). I checked all the plants in operation and only one is by ExxonMobil - listed as "Shute Creek Gas Processing Plant" with a capture capacity of 7.0 million metric tons. Thus, ExxonMobil captures 7.0/33.3 * 100%, or just 21% of all captured CO2 annually. However, ExxonMobil's claims are "cumulative." Performing similar math with the same website for cumulative CO2 captured gives 265.1 million metric tons of CO2 not including Exxon's plant. To capture 40% of all cumulative CO2, ExxonMobil would need to have captured 176.733 million metric tons (again, check my math please). Assuming that the ExxonMobil plant captured 4 Mt/year from 1986 to 2008, closed 2 years for expansion, and opened from 2010 to the present day and capturing 7 Mt/year, ExxonMobil's lone plant would have captured 155 million metric tons of CO2. This is equivalent to 36.9% and is not enough to be "more than 40 percent of cumulative CO2 captured."
Speculation on 3: Performing similar calculations of MtCO2 captured, but per company, shows that each has captured less than 132 MtCO2. Thus, ExxonMobil has "cumulatively captured more CO2 than any other company." The assumption is that (1) all CO2 capturing plants operated by a company are listed on the document, (2) that all capacity values (MtCO2/year) are accurate, and (3) the ExxonMobil plant captures carbon at an average of 4 MtCO2/year. Note that I did not assume an average rate of 7 MtCO2/year. This is because of a 2010 plant expansion. I used 4 MtCO2/year because MIT lists the "start date" as "1986. 2008 (4 Mt/yr); 2010 (6 Mt/yr)." This isn't absolutely clear, but appears to indicate that 4 Mt/yr was captured between 1986 and 2008 before the plant expansion.
Regarding 3: Using the same site as for 2, and performing individual calculations for each company, I confirmed that ExxonMobil has "cumulatively captured more CO2 than any other company." The runner up is Occidental Petroleum and Sandridge Energy at (currently) 75.6 million metric tons of CO2.
Does ExxonMobil capture "about 7 million tonnes per year of CO2?"
Yes, ExxonMobil captures "about 7 million tonnes per year of CO2."
Has ExxonMobil "cumulatively captured more CO2 than any other company?"
Yes, ExxonMobil has "cumulatively captured more CO2 than any other company."
Is ExxonMobil responsible for "more than 40 percent of cumulative CO2 captured?"
No, ExxonMobil is NOT responsible for "more than 40 percent of cumulative CO2 captured." ExxonMobil is responsible for 36.9 percent of cumulative CO2 captured.
As @DevSolar says ExxonMobil isn't exactly "'capturing for storage', they are using an unwanted byproduct of fossil fuel extraction to ease further extraction, i.e. getting at more oil (which will eventually be turned into CO2)."
Note: I have not found other ExxonMobil CO2 capture plants that have shut down and thus, aren't in the Global CCS Institute's database. Please comment if you do.