Does the track have an on site hobby shop?
If so, then I would encourage you to get a kit from them where they will stock the parts you need.
If they don't have a shop and you will need to support yourself for parts, then I would consider
PR Racing, they are relatively new to the US market and offer high quality kits for extremely reasonably pricing.
Is your budget $350 for just the kit or is that figure for all the electronics too?
You say your battery/charger are a few years old... what type of battery is it? Most modern turf based buggies require Shorty LiPo packs these days and chances are your battery is long gone if it hasn't been used in over a year.
If I was going to build a car for a family member today who is on a tight budget, this is what I would get:
$168 PR S1 V3 FM Sport
$80 Radio (I have a Radiolink RC4G that is practically new that I will sell for $50 shipped with 2 Rx's)
$82 17.5T Motor/ESC Combo
$35 2S Shorty LiPo
$40 Servo (I have a decent servo that I will sell for $20 if you buy the radio from me)
$10 Rear Tires
$3 Rear Inserts
$10 Front Tires (no foams needed)
Rough Total = $450 after shipping and accessories for paint/glue/diff/shock fluids