Yes.
My games archive

Does the site keep a record of my games against the computer? (Play > Vs. Computer) ... I can't find any record of those games on the website.
Yes.
Where are they, and are they accessible to me?
I go to Learn > Game Explorer, and choose My Games from the menu. The only options there are Online (correspondence) and Live.
If I go to Home > Game Archive and choose Most Recent Games from the menu, nothing is returned when I search. The only games I've played are against the computer.
Are other people able to see their history of games against the computer? (This is not Live Chess with a computer opponent, but rather, the Play > Vs. Computer option that I mentioned in the original post).
I think NM-Dale thought you meant against the Live Chess computers, rather than the Play Against The Computer computers, which are different.

I'd like this option to be available too.
I played a match with the computer and the app crashed after the game so I didn't have time to note/analyse the game. That's so unfortunate.

I'd like this option to be available too.
I played a match with the computer and the app crashed after the game so I didn't have time to note/analyse the game. That's so unfortunate.
The site has added a Save option, which will be coming to the apps, if they don't already have it, but it is manual and wouldn't help you if the app crashed.
Does the site keep a record of my games against the computer? (Play > Vs. Computer)
At the moment, I just copy and keep hold of the PGN for future analysis, because I can't find any record of those games on the website, but it would be nice to be able to see a record of those, just like with live games.