No, they did not.
Frequency hopping was not only known but was in use decades before Ms. Lamarr and Mr. Antheil came up with a different way to do it. Though they were rightly given a patent for an interesting way to do frequency hopping they most certainly were not its inventors. The technologies we use today would be here just as they are now had the inventors never been born.
Jonathan Zenneck's book Wireless Telegraphy (German, 1908, English translation McGraw Hill, 1915) is one of the first public documents mentioning frequency hopping - a full 34 years before Lamarr and Antheil's patent. And it has been written that Germany was using this technology in 1918 against the British (see the EE Times article referenced below.)
Patent #US723188 by none other than Nikolai Tesla in 1903 does not use the phrase "frequency hopping" though it describes changing wireless frequencies to avoid interception.
Patent #US1869659 by Willem Broertjes submitted in 1929 and issued in 1932 (a decade before Lamarr and Antheil's) describes what would become known as frequency hopping spread spectrum:
The essential feature of the invention resides in the fact that messages are transmitted by means of a group of frequencies (working frequencies) known to the sender and receiver alone, and alternated at will during transmission of the messages.
Lastly, here is an article from EE Times noting their accomplishment but demonstrating they did not invent the technology: A short history of spread spectrum says,
Nikola Tesla, the prolific Serbian-American inventor and radio pioneer, filed a U.S. patent, granted on March 17, 1903 which doesn’t mention the phrase “frequency hopping” directly, but certainly alludes to it.
Such an interesting idea didn’t escape the military’s attention of course, and by 1915, the Germans were making use of primitive frequency hopping radio to stop the British eavesdropping on their conversations.
Lamarr, together with co-inventor George Antheil, a pianist and Hollywood composer, came up with a system for radio control of torpedoes. The idea was not new, but Lamarr’s concept of frequency hopping to prevent the intended target from jamming the controller’s transmissions was.
While the concept of frequency hopping is used in spread spectrum communications there is no evidence either inventor had a role in SIGSALY or any other spread spectrum project.
Their patent #US2202387 does not discuss spread spectrum in any way.
Personally, my first exposure to frequency hopping was the RT-1209 military HF radio in 1988. It used frequency hopping to prevent detection, interception, and resistance to jamming. It used a preset set of frequencies to 'hop' between. This sequence was determined using a 'time of day' word that was programmed into the radios wishing to communicate.
WiFi on the other hand did not come into existence until 1998 - more than a decade after the military radio had already been fielded.
Lastly, this is not to diminish Ms. Lamarr - she did much to be proud of and it should be acknowledged. Instead this is meant to diminish the erroneous claims made by others.