M.D.
I have a success story for you!
My older (now 11) had mild tongue thrust plus an overbite (although not an open bite) and the lisp on S and Z sounds. We took him to an orthodontic consult. My son already had his adult 2 front teeth in and they were pointed at an angle out with a small gap between them. According to the ortho, it was a bit of a self-propagating loop. As the teeth came in, there was nothing to hold the tongue back so the tongue pushed the teeth. Then because the teeth were pushed, there was still nothing to hold the tongue back so it pushed the teeth more. It also lead to the lisp on S and z sounds, especially when they were in the middle of words and harder to control.
The ortho suggested braces to pull the teeth back and get rid of the gap, followed by a a bionator to correct the overbite, and then retainer to keep the the teeth in place. We did it, and it worked. The braces were only for about 7 months. Then he was in the bionator 24/7 for about 6 months, then bionator only while sleeping for 6 months, and now he's been in regular clear plastic retainers (top and bottom) while sleeping for about 6 months. He'll stay in the retainers while sleeping until all his permanent teeth are in, to guide them into place as they come in. The front teeth are back in place and the lisp completely resolved on its own once the front teeth were in place. They will make a new retainer every 6 months or so since he's constantly got teeth falling out and growing in at this age. They do not think that he's going to need a "phase 2" of braces. It might sound like a lot, but only 6 months of braces is really short, and the rest of it is all very easily - mostly just bionator/retainer while sleeping.
We didn't go for the consult until he was almost 9, and the ortho put us straight into the treatment plan - I think he would have done it sooner if we had gone sooner.
Good luck - and make sure you get a few ortho consults. We got 2, and one practice wanted to do much more extensive stuff than the other. We went with the less invasive plan and are very happy.
ETA: We did get an extensive speech eval at a center that does all kinds of developmental testing for kids. The speech therapist we saw said that in our son's case, she wanted him to start the ortho work first, then come back for a re-evaluation if the ortho work did not fix the lisp. Your daughter's case might be different from my son's, so I think that going for the speech eval is a good idea. But your child is also old enough for the ortho consult, and you might get the best results if you address both issues with professionals (speech therapist and ortho) together.