OK. Somehow I had the impression that the story was more about post-reunion stuff.
So he has to find her, not the other way around (which would be easier, because his name would not change). And it has to be deliberate, not accidental. And it has to be tough, not fall into his lap. So she can't be at the same school as him. This makes it tougher because you have to find a way to overlap different groups of kids.
I assume she's in the same town/ city as he is, otherwise there's not going to be a story since 10 year-olds can't just go off traveling to hunt for a sister. And he can't drive. At age 10, it's unlikely he'll be permitted to take public transport independently, even if it exists in the town or city where he lives. 1980s, so no internet.
Does he have adult assistance in this?
If he does, then there's a chance. He knows her birthdate. Since she'd have been 4 also, there's a good chance that her a-parents wouldn't have changed her given name, or would have retained her original name as her middle name. So if it's a really unusual name, there's a possibility.
If not, here's one possible route. Since he knows her birthday...he sort of networks through friends. It's useful that he's got a girl ally, because at 10 boys and girls are often running in different circles.
So the friend (lets posit she's an active, extrovert girl) gets the word out to all the girls in her church camp/ sports camp/ after-school ballet class that she's looking for someone adopted, with a 12 December birthday, in fifth grade, named Praxitelia. One girl says she has a friend with that birthday, but she's called Emily P. So your protag and pal decide to try contacting Emily P. They phone her; she's not the one, but there's another girl she knows with the same birthday...
And so on. The problem with this is it makes the friend the main actor, but you really want the effort to come from the protag.
Incidentally, the fact they're twins makes it more likely that they'd want to keep them together. Something went wrong in the system, but you don't have to explain what. You can handwave your way past it. Whatever it was, the kids won't know, and perhaps no one would without an investigation. Things go wrong in the foster care system frequently enough for it to be plausible.