I am looking for someone to record a few children's songs for me (for
a children's play gym).

These are short 30 to 60 second songs to the tunes of Twinkle,
Twinkle; You are My Sunshine, etc, with custom lyrics we have written.

I'm looking to contract someone who can produce this in a professional
sounding way, with voice and optionally instrumental backup.  I'd like
a Raffi or Wiggles or Sugar Beats type of a sound.

A successful answer to this question would include links/contact
information for 3 people / groups who are in the business of doing
this sort of thing on a contract basis, and have some experience with
kids type music.  I don't want jingle production companies that are in
the business of writing their own lyrics/music for commercials, unless
they also do this.

Do the singers have to be within a certain geographical area?

And also, what kind of budget do you have for this? And what kind of
deadline are you on, if any?

the geographic location would definitely help.

I would check www.craigslist.com search the local area.