Next week, I want to call a radio station for a contest. The phone
lines will be jammed by others trying to get through. I'd like to find
a service where, on a one-time basis, I can pay some reasonable amount
(less than $20, I'd think) and they call the number repeatedly, then
hook me in when they get the radio station's phone to ring. Not
looking for equipment, long-term commitments, etc. Just a one-time
offering to call a number repeatedly until it gets through. Thanks for your help!

Some fax machines will redial over and over when they get a busy
signal. If you have such a fax machine, you could set it to dial the
number, then pick up the handset if the call goes through.

Another way, though tedious, is to dial the number.  Immediately
engaged signal is heard, disconnect then get the dialling tone again
and hit the 'Redial' button.  Keep doing this until, hopefully, you
may get through.  Even if you do, you may not get into the studio!

Can't some mobile phones do 'automatic redial'?

What kind of number are you calling?  If it is a 900 pay-by-call
number and you get an answer that all lines are busy (not just a busy
signal), the station is most happy if people just keep calling again
and again.