Most of our cars are Tri-cars so have one drive wheel in the rear and two steering wheels in the front and most of our races are tight parking lot races with lots of turns.
So for the rear we use 20" BMX rear wheels. I buy mostly 48 spoke, double wall rims and sealed bearings. These are a little more costly but they are tougher and tend not to break as many spokes and not taco. Just be sure that you tighten the spokes before every race. Lighter and plastic wheels tend to break during a race because of side loads.
Wheels for the front also need to be very tough as they take on a lot of side load and most bike wheels will not work on a competitive car that take corners has high speeds. These wheels also need to be the braking wheels so must hold a break. We have used Tomos Moped wheels for years. They are a little heavy but are easy to work with and will hold up with our abuse. They also come with a fairly good set of bearings that have a 12mm i.d. and have drum brakes that work well and are easy for my kids to keep functioning. Disc brakes are great but on rough tracks of if not handled very gently they tend to warp or bend and need replacement and they are also expensive.
For our tricycle cars we use BMX wheels for all three. We usually make our own caliper breaks for the rear wheels using purchased pads but with the homemade system you can add more force (stopping power) to make them work effectively.
We use 20" bike because you can buy very high quality ones. Our rear 20" bike wheels are double wall with sealed bearings and 48 spokes that are laced to help with side loads. We use Tomos Moped wheels on the fronts which are aluminum mag wheels with an included drum brake so are nearly bullet proof and very easy to use. The 16" do not have has much side torque so should not taco as much on corners but most have less quality and are not double rim and also have poorer bearings. We tried some cheap normal bike store 16" rims and they did not last a full season of tight track with the speeds we travel. If I was using bike wheels on the front of my car I would try to find the best 16" I could because front wheels take so much lateral force on corners. We just use bikes on the rear drive wheel for any of our tri-cars designed for tight tracks. We have a couple of tricycle style and those we use 20" on one a 16" on the other bike wheels. But again I went to a local bike manufacturer (Bike Friday) that makes high end bikes with 16" wheels and had them sell me a set of top shelf rims and they work great. One of our tricycles had cheaper 20" rims on the rear and both of the bearings on one rear wheel disintegrated this last weekend.
Here in Florida most of our races are run on tight parking lot courses, too. I use 48 spoke BMX wheels on the rear (same as Mike above). After a few races, when they start breaking spokes, I re-lace them with 12 gauge spokes.
For the fronts I build a sort of "hybrid" wheel. I use a sealed bearing 36 hole mountain bike hub that accepts a standard 6-bolt brake disc and has a 15mm thru-axle center. (I would use 48 spoke, but they are not available!) I use double-wall alloy rims (Weinmann DM-30 is a good one for the price) and lace them with 12 gauge spokes. I check and adjust spoke tension before every race. Here in Florida a race day consists of two 1-hour races run back-to-back with a break between (usually 90 minutes to 2 hours). I also check for loose or broken spokes between the races and tighten or replace as necessary. After a few years in Electrathon you get to be pretty quick and efficient at replacing spokes... I build my cars with 7 degrees negative camber in the front end so the stress is reduced slightly on the outside wheel in a corner.
I use Bengal brand disc brakes. They are very effective, durable, and easy to adjust. I have also used MTB brakes in the past - they are good durable quality, but more temperamental to adjustment
I have laced a set of 16" wheels using the same hubs I described above. They are 36 spoke hubs and rims, but the rims are single-wall because that's all I could find. Also, I had to use 14 gauge spokes because nobody makes a 12 gauge spoke in the correct length. I have not tried them out yet because my current cars are set up for 20". I'm hoping that the decreased wheel diameter will lessen the stress on the spokes and rim so that they will hold up. If/when I have test results I will post them here.