One of my guitars is a Mexican Fender Jaguar that still has the stock vintage style tuners on it (I've had the guitar around 4 months by now). The tuners have been fine up until the last 2 or 3 string changes (the problem started this month).

I've had a lot of trouble getting the 1st and 2nd strings in tune the last 2 or 3 times. It's mostly the 1st string, though. It can barely get up to the right pitch and I have to fight it to get there (it just instantly drops down in pitch, the closest it stays to 330 Hz is 20 cents below). The 2nd string has been better and I've only had trouble with it today with the string change where it stayed at 12 cents below the right pitch, then hovered at -5 for a while - I eventually got it up to the right pitch. This may have been a case of needing to stretch it some more, but the stretching process took much longer than it usually does for me. I happened to do my string change today and both high e strings I tried broke (neither stayed in tune at 330 Hz). I'm not awful at string changing, by the way, this is the first time I've broken a string in a long time and it happened to be 2 high e strings...

The washer for the high e looks to be on fine (it's not loose) and the back of the tuner seems to be in just fine, too, but the actual machine head where the string goes wobbles and moves quite a lot in all directions. I haven't noticed this in the other machine heads, especially not the thicker strings. About 2 months ago, the gears for the 2nd string tuner were quite stiff so I lubricated them and haven't had a problem with that machine head since but it might be worth mentioning as this set of tuners has had a problem here and there before.

In terms of my restringing process, I always do stretch my strings until they stay in tune. Once or twice, the strings (1st and 2nd) have caught either in the saddle or nut (I suspect the nut) but have tuned up fine after I've 'freed' them. This has only really happened since using D'addario strings, I'm not sure if that's coincidence and bad timing for the D'addario strings that I decided to try or not. I know the guitar's been set up properly for my playing (I play 9 gauge strings), so I don't think it's a case of bad set up...

Ultimately, I'm trying to decide if getting vintage style *locking* tuners will be a good option or not. The tuners I have now are not the greatest, I know, as they're stock and they've had a few problems before but new tuners are quite expensive and I want to make sure they are a good investment for my problem... If new tuners seem like a good option, are the Fender vintage style locking tuners a good buy or not?

Hopefully the information I've provided is fairly clear... Thanks to anyone who can help!
sounds like the 1 tuner may be loose. check that first. the nut is often the real culprit in tuning issues but it sounds like you may have a loose or bad tuner in this case
I just posted this in another thread.

Several things can cause tuning problems, but this time it seems to be probably the tuners. So...

Lubrication - One drop of 3 in 1 oil about once a year will keep them in good shape. One drop. If they have a hole in the back side, oil there, if not, let a drop dribble down the tuner post from the string side.

Grommets or nuts on the string side...make sure the nuts are snug, not too tight. Grommets, can't do much with them but make sure they are seated well. Tap lightly with a screwdriver. Hold it in place with one hand, bump it with the other hand, lightly.

Nut slots - Strings can bind in the nut slots, use a little graphite from a #2 pencil, just scrape it through each nut slot.

Ball ends - I've gotten strings with badly twisted ball ends, tuning would go downhill quick until one night a string came loose at the ball end. For several years while I played for a living, I soldered the ball ends to stop this. I played 5 and 6 nights a week, changed strings every night. soldered ball ends every day.

Tuning - Always tune up to the pitch you want, never down. Start below pitch, even if you have to de tune down, then tune UP to the proper pitch. If you tune down and leave it, you have a good chance the string can loosen the first time you stretch it, and it will go flat.

I've never seen a need for locking tuners, if you tune properly, they aren't needed. I have a 1966 Harmony and a 1940 to 1950 lap steel, both with the old brass open back tuners used a lot on cheap beginner guitars in the 60's, both stay in tune quite well if I tune properly. Cheapest tuners ever made, one set is over 50 years old, the other about 75 years old, both still work great. I just use a drop of oil once a year or so and tune properly.

If your tuners are lubed and still tight, loosen the nut on the tuner post, it might be too tight. Then always tune up, not down, even funky tuners should stay in tune well.
Hmmm...I wonder what this button does...