Fellow singer here. There are a few approaches.
Approach 1: learn the tunes in standard keys from the various Real Books (get yours in C), but ff a song is too high or too low, create your own takes on the melodies that go into the registers you need, so that you can swing it without a lot of hassle. Just like playing an instrument, you should be able to use a combination of the chord progression and the written melody to get around a chart. Please note that some tunes (like Autumn Leaves) might be in a couple of standard keys, so you might get Gm one night and Am the next.
Approach 2: If you have figured out that your best key is something other than in the Real Book, be ready to provide charts in your key to the band. But make sure that the charts are as close to the look/format of the Real Book as you can. Use MuseScore or another engraving program to make your charts nice. Include melody, lyrics, and chord symbols (use Marker Felt or a similar font for those).
Either approach should be supported with having your own book or binder with you, with multiple copies of the tunes you'd like to do, in case any of the players has an incomplete book. Depending on the venue's resources, you may consider having a simple cheap mic and a little guitar amp in the trunk. I had to do this at a restaurant in San Francisco when I'd join in with the band. The place didn't have a P.A.
You should have an idea about what you'd like to do musically, and be able to nicely describe it (e.g. on 'Round Midnight or lots of other tunes: "Can we start it as a ballad, but kick it up to a swing after the second ending?") This would be a super normal way to go. Or you could do a ballad like I'm Old Fashioned straight ahead as a hard bop tune--if you've worked it out, experienced players will be supportive. As you build a repertoire, be judicious about what you add to your book--don't halfway learn 20 tunes--learn to sing 5-10 tunes really well, and be able to solo over them.
Just be prepared for what comes at you! I was sitting in at the old Blue Wisp in Cincy when I was in college, blowing over All Blues. At one point, the band (all pros) went double time under my scat solo, and it was amazing--took us to another planet of expression, and since I didn't balk, I was invited back. Another time I was sitting in at Birdland, and I got lost somewhere in my 3rd scat chorus over My Funny Valentine. At this point, I pointed to my noggin to get us back to the head, and the band went there so we could finish with dignity intact.
I've had bandleaders yell "you're not done yet!" at me when I thought I'd duck out after one solo chorus. You just have to get up there and try it.
