First the words "budget" and "race car" should never be used together. It's an oxymoron.
That said. How serious of a race car are you thinking of building? There are several things you can do to really make the Foxbody a great racecar that don't cost a lot of money.
Power is great, but the chassis needs to be addressed long before power is introduced.
If I were to build a race car, I'd build it like so.....
1.) Find the cleanest, rust free, 4 cylinder coupe I could.
2.) I'd strip the car to a bare shell.
3.) I'd sell NOTHING off it until I finished building the car.
Never know what you might need.
4.) All Foxbody are spot welded together. Grind and strip those seams and stitch weld them up. I would make a jig to hold the car square and level when I did this. Why? You don't want to stitch weld the car crooked. keep it square and flat and it will be easier to make it go straight down the track.
5.) I'd spend $1500 on Race
suspension (for example.....the front and rear packages that UPR sells)
6.) Put a roll cage in it that exceed the requirement for your planned fastest ET.
7.) I susbscribe to the thought process of if it doesn't make it go faster, it doesn't need it. Sound deadening, don't need it. Back seat, don't need it. But, of course, consult the rules for the sancioning body you will race under. if you are gonna race just the local track, check them, if you are gonna run NMRA, check with them. They WILL have different rules. NMRA, for example, requires the interior to look stock, while the local track won't likely care.
8.) I wouldn't seriously race with anything less than a 9", so I'd buy one of those too.
9.) Now it's time for an engine. What kind of power do you want? Do you want something that you have to tinker with or something that's reliable and fun and requires minimal maintenance? Sure that custom EFI twin turbo, nitrous 302 sounds like a cool thing to play with, but what's the reliability and maintenance gonna be like? What's the reliability and maintence of a mild 552 big block gonna be like?
10.) Now what kind of transmission do you want? manual or auto?
This is a lot more involved than you may have thought. And if this was gonna be for a job to earn extra income (BWAHH HA HA HA HA.....sorry), it'd be way worse.