Is this a youth league or adult league? If it's adult league then what Richie says is true, it would have to be a league/tournament rule. USBC has ruled on this several years ago. As long as the bowler is using the same dominant hand for the swing and release it is legal to go from one handed to two handed and back. In the online rule book at bowl.com, it can be found in rule 118b-Commonly Asked Questions. It is 118b/3.
Rule 118b also states: A bowler using a two-handed delivery will establish an average with two hands. When the bowler changes to a one-handed delivery, they must establish a new average using that hand.
What if the person switches for spares?
I think the spirit of the rule for people to not sandbag, I guess.
You can switch to one handed for spares and vice versa as long as the ball is in the same dominant hand. USBC release a statement about that on the bowl.com website either last year or the year before. That is for adults, I believe in youth leagues/tournaments they have to use one or the other.
When your in a scratch tourney, league, or association you can use whatever hand or hands you want as there is no handicap adjustment involved.
I believe that is false, as handicap isn't the reason for the rule. Averages are still maintained and all USBC sanctioned averages have to be maintained using either one hand or the other and no combination of both.