There's no rule that says a half-blood can't be adopted, but there's no provision saying they can, either. The rules sort of assume no one would want to.
It would probably be best if they just ask to foster her for now. Finnigan's foster father really acts like his father; if you CAN adopt your foster child he'll do it and once he does, there'll be a rule saying you can. Because Mr Rosier is part of the same inner circle as Mr Malfoy and Mr Nott and the rest. If he wants something he'll get it.
Re: I Solemnly Swear That I Am Up to No Good
It would probably be best if they just ask to foster her for now. Finnigan's foster father really acts like his father; if you CAN adopt your foster child he'll do it and once he does, there'll be a rule saying you can. Because Mr Rosier is part of the same inner circle as Mr Malfoy and Mr Nott and the rest. If he wants something he'll get it.