I am considering changing my amp, It is a JVC AX700 that I purchased new in the late 80's.It powers a Rega P2 TT with a Rega fono stage through JBL L96 speakers and a B&W sub. I am considering changing to either another solid state or tube amp used in the $500-$1000 range. Is it realistic to improve sound quality with these parameters? Should I stay with an integrated amp or pre-amp/power amp? Any recommendations for a change?
The answer to your original question might be "maybe". I'd say more like yes, of course! You asked about replacing the amp and then it seems you get sensitive about criticism of the JVC. That's to be expected if you've owned it since new, but you asked if replacing it would be an improvement.
I have to wonder how many here who have suggested your problem is with the speakers have spent much time with the L96. I have a mint pair, as well as over a dozen JBLs from the L1 to the 4345. I consider the L96 one pair that won't leave my home until I'm dead and gone (probably the not-so-distant future for those of you into a good garage sale, but my wife will have a list of values by then!

) I've often said they don't do anything badly and do many things so well they would be my "desert-island" speaker if I could only keep one pair and they had to fit in any room situation.
The L96 may seldom do any wrong but they will improve with more power added. I've run them with up to 200 watts-per-channel of Crown power. I'm not sure how that will compare on your JVC's 115-wpc, but let's just say not all claimed watts are equal. There is also a certain synergy between amp and speakers that can't be determined until you've tried it. Since your basic supposition is that you'll be replacing the JVC, then I'd start with separates which will give you the ability to try different power amps. Personally I've used Soundcraftsmen and Crown products for decades, many since new, and stick with them because of the way they sound and how they've served me. Many will tell you Crown amps and JBL speakers go together. Also separates will give you the ability to consider powered speakers. I also own two pairs of powered JBL LSR305s and run them off a Crown PSL-2. They are the most impressive
affordable modern JBLs I've ever heard.
My daily-drivers are a pair of bi-amped JBL 4345 so I know what an impressive speaker sounds like, and I'm still impressed with the L96 which I can compare in my home to the L112, 4412A, LSR305, L3, L5, L7, and others in the same room with a variety of amplifiers from Crown, Adcom, Carver, JBL/Urei, and Soundcraftsmen. You don't say where you live but maybe there's an AK member nearby willing to let you try some gear before you make the jump. I only have one amp I've paid over $200 for (not saying I didn't get a good deal or they're not worth much, much more . . . ) and I have one currently out on loan to a neighbor. I have pre-amps I've paid over double that for and some that fall well within your budget. I think the best bang-for-the-buck comes from vintage gear. I've heard arguments both ways so everyone can save their breath.
Personally, I'd go for a vintage pre-amp which incorporates a good phone section. The Crown PSL-2 is excellent if a bit pricey. I have two since I can't pass up a good deal on a good product I've enjoyed. But my main system uses a Soundcraftsmen Pro-Control-Four with a Pro-EQ 44 since I love having an EQ in every room to adjust for low-volume listening and room issues. If you like matching components you could do a Soundcraftsmen Pro-Control-Four and Pro-Power Four even including the great Pro-EQ 44 for well within your budget. With Crown you could do the PSL-2 or SL2 with a PS-400 amp also within your budget. Those have all served me well and are what power my two main systems today (though the SC amps are sitting idle and I use Crowns in all systems currently).
So, yes, I'd say you'd benefit from a new amp, but go with separates and see if you can try a few in your system first before puling the trigger. Yes, your JBL L96s are fine speakers, but there are others out there that might serve you better depending on your musical tastes, etc. I wouldn't consider running a sub with mine but I'm not expecting them to give a home-theater experience, either. Give it a try and use your own ears to make your own decisions. That's what really matters.