There does not seem to be any supporting evidence for the claim.
By 'vegetable oils' they must mean 'seed oils', which have been the object of discussion among the health-conscious.
The specific claim made in the short video has been examined by Insider in their article Some keto evangelists believe vegetable oil is worse than cigarettes, but the science behind the theory doesn't add up. It mentions Ben Azadi, founder of the health coaching site Keto Kamp, as seen in the video.
There's little peer-reviewed research to back up claims that seed oil is worse than smoking
While there's evidence processed foods and food fried in oil aren't good for our health, claims about the risks of seed oil in particular go far beyond what current research can support. Azadi said his comments are based on the work of Dr. Cate Shanahan, a family physician who writes frequently about vegetables oils. Many of the studies she cites suggest correlation, not causation, between oils and illness, or focus on lab rodents, not humans.
Why fears of inflammation are also over-hyped
Among keto evangelists, though, there has been some concern that too much omega-6 from seed oils, and too little omega-3 (found in greater concentration in things like fatty fish), can cause inflammation in the body. That's because one type of omega-6, arachidonic acid, is linked to molecules that promote inflammation.
Research has suggested that's not the case — studies have shown it does not increase inflammation, and in somes cases reduces it. Although it's not entirely clear how it works, it appears that arachidonic acid can also be converted by the body into molecules that help fight inflammation.
That Research links to the National Library of Medicine, NCBI Literature Resources.
Omega-6 fatty acids and inflammation
... Hence, it is commonly believed that increasing dietary intake of the omega-6 fatty acids ARA or its precursor linoleic acid (LA) will increase inflammation. However, studies in healthy human adults have found that increased intake of ARA or LA does not increase the concentrations of many inflammatory markers. Epidemiological studies have even suggested that ARA and LA may be linked to reduced inflammation. Contrastingly, there is also evidence that a high omega-6 fatty acid diet inhibits the anti-inflammatory and inflammation-resolving effect of the omega-3 fatty acids. Thus, the interaction of omega-3 and omega-6 fatty acids and their lipid mediators in the context of inflammation is complex and still not properly understood.
The Insider also mentions trans fats:
A now-banned version of seed oil was linked to inflammation
The evidence on the health risks of trans fats is so substantial that trans fats and oils that contain them have been banned in the US since 2018.
Perhaps the keto diet activists haven't updated since then.
The video makes an unfair comparison. Although the text caption says
Smoking cancer risk = 16%. Vegetable oils cancer risk = 86%
the narrator slips heart disease into it: comparing death from cancer among smokers with cancer and heart disease among those who consume vegetable oils. Wikipedia shows that amongst the whole population heart disease and cancer are the top two causes of death, with heart disease at the top – not cancer. On these grounds alone, the narrator Ben Azadi has twisted the case.
But anyway, there is a table for the US in 2020 given by Medical News Today, and it states
the 10 leading causes of death in the U.S. accounted for 74.1% of the total deaths
These top ten together are less than the 86% claimed, which given that almost everybody has some daily intake of seed oils, seems to be wholly spurious: you might as well claim it is due to drinking water.
I have so far avoided reference to Brian Peskin, who the narrator Ben Azadi cites as his source. According to QuackWatch in 2002 he was fined by the State of Texas for making unsubstantiated claims about Radiant Health Products and Peskin’s credentials.