Not sure what you are closing on. If a new purchaser from Disney and you have a contract, you can make that reservation now. If a resale purchase, you won't be able to make a ressie until you are in Disney's system which will not occur until sometime after the closing (perhaps a week to two) .
Decent chance you will be able to reserve time you want in June but possibly not at resort that is your first choice. For June, except for time in the end of the month that takes you close to the week in which July 4 is in, it is usually not overly difficult to get something even when reserving six months out (e.g., OKW, SSR, or BWV preferred pool/garden view are more likely than others to be available)