I did tennis LTA tournaments at low level, and assuming it hasn't changed:
You're looking for the 10 and under competitions (it's age on 1st January that counts, so if she's had her birthday then she'll still be 10 and under next year).
You will need to find out the tournament referee and apply some time in advance. they may only take the first certain number (often not many in under 10s, perhaps 16) and it will be first come first served. You will pay for this as you apply, if you don't go, then you lose the money.
In under 10s it's often round robin (everyone plays each otherand sometimes those results are used for a knockout competition so they got lots of matches)
You will be required to be available all the tournament time until you're knocked out. You will receive a message telling you the time of the first match, and after that you're assumed to be available. You generally can't say you're busy (one person was knocked out for attending her dad's funeral in my first tournament) Even if it's raining you have to wait through it.
Finals are usually held on the last day and prizes (trophy for winner, runner up and possible a small cash prize-I am talking small, often not more than the entry fee) presented. You won't make a profit that's for sure. I won all tournaments entered one year, and if you took petrol into consideration (but not equipment) it didn't cover it.
You need to make sure she can do her own calling (out/in) and keep score. It is entirely up to her (not you) when she's on court. You don't interfere at all even if the other person's a pain. You can appeal to the tournament referee and, if he thinks you have a case, then he can send someone to umpire.
And be prepared for her to lose. Sometimes a lot. When I played it wasn't infrequent that someone turned up to a tournament thinking they were the next best thing, and got completely knocked off court. I remember one saying to me in bewilderment "but my coach said I was good", difficult to answer when they've been thoroughly beaten by everyone. Some children thrive in the competition, some sink. The first season often lose quite a bit simply because they're getting used to it.
They may need to play a number of matches on a day, being too tired isn't an excuse for not playing. In round Robins (often only best of 7 games or 1 set or something) they may play the entire tournament in one day. If playing the best of three sets, they can be asked to play 2 full matches in a day with only half an hour between them. I came off once after a 3 hour match and had only forty minutes before I went back on
And there will be some overly competitive parents (remember one pair who used to sit either side of the court keeping a running commentry on their car phones through the whole match-that was in the late 80s!) and some nasty children (one at our club was known as Brat Mac for very good reason!)