I have a Vista 10 speed conversion, circa late 70s early 80s. The bottom bracket has a threadless spindle design: two cups that hold the bearings in, spindle "pegs" that attach to the cranks. Fair enough. The problem here is that they keep gradually loosening over time. I took it apart, regreased everything, put it back together. It ran fine for a ride or two but it's loose again. There is no lock ring

I'm a bit confused by the lack of a lock ring, nothing was lost in the conversion. Which means: the bike either didn't have a lock ring when I got it (it was lost) or the bottom bracket never had a lock ring. The former seems more likely (I didn't loose anything).

Anyway, I have a new chainring/crankset I wanted to put on there anyway but the bottom bracket on the frame isn't threaded. Are there any new square taper bottom brackets that this frame type will accept?

if its standard threading (ISO) and 68 mm then you can get just about any bb, i would suggest a shimano square taper one because its only about 20 bucks. before you do this i would suggest just getting a BB lockring, you can probably find one for a few dollars at your LBS

Fail.

It sounds like a cottered BB to me, is it?

To the best of my knowledge it's a "cup and cone" style bottom bracket.

Ha, I got a good laugh out of your reply to the above post.

So, you have the tools to install the bb because you said you took it apart and regreased it. If the BB is in good shape, why don't you just go buy a lockring and put it on?

You have to explain this more. What threads are youtalking about?