If I have a petty theft charge that occurred when I was a juvenile, will it show up on a livescan report?

If you were fingerprinted they it may show up. You need to petition the DOJ to seal this if it does.

It all depends on the underlying felony that you where convicted of. Need more info.

What can I do if I pleaded no contest in court?

Finish the terms of your summary probation and ask for PC 1203.4 expungement.