How slow is "slow"? Being new to Lemons, you're going to be slow no matter what you're driving. Just ask Robert and Roger about my first time driving many years ago -- CMP short course with ~120 cars... I was dog-ass-slow, driving like I'm on the interstate. I don't think I ever took the engine over 4000 rpm. "Apparently that's as fast as he wants to go. He's not breaking the car, or getting any flags, so leave him out there."
Don't fret over the budget. Don't go insane with obvious "race" components -- they won't help anyway. Make the car safe, stable, and reliable. As you get more and more experience, you may find that speed isn't all that necessary. "to win the race, one must first finish the race" Just staying on track, turning laps, it's surprising how high in the standing you can end up. Winning, in any class, requires a strategic amount of luck. We lucked up at Sebring... nobody was there. We lucked up big time at Barber... the 5 cars beating us all broke and Craig (Terminally Confused Del Sol) was doing pit stops at the pumps which kept him half a dozen laps behind us. (and he really didn't want to win and end up in class A forever. which is why we no longer race the civic.)
Aside from the obvious things to get it through tech -- cage, seat, harness, windshield, hood pins, fire extinguisher, etc. -- the first / only "performance" modification I would highly recommend is upgrading the brakes: suitable "race" pads, and high temp brake fluid (ATE and Pentosin are great and reasonably priced. And Pentosin is likely what VW put in it in the first place.) Not to start a brake pad debate, I've used EBC yellow stuff on my bug ('01 1.8T) fronts, and Pep Boys prostop ceramics on the rear as I don't turn of ABS or ASR and it can't deal with the high friction rear pads. Without a proportioning valve, the fronts do all the work anyway. And EBC slotted rotors, front and rear.
Duff Beer Civic (#128) -- 2014 Sebring - Class B (#1 of 7), 2016 Barber - Class B
1981 Jet Electrica 007 [Plymouth Horizon TC3] (#128) -- Mk.1 - Index of Effluency Eco (IOEe) @ 2016 Lemons South Fall, Mk.2 - Judges' Choice @ 2017 'Shine Country Classic, Mk.3 - Index of Effluency @ 2017 Southern Discomfort