Grace & Peace!
I don’t know why you would, but I also don’t know that you shouldn’t. It seems like it would be perfectly licit for you to wear cassock and alb or, as in the Anglican office, the proper choir dress (e.g. academic hood, surplice, cassock).
But I don’t know what you gain by it. The temptation, it seems to me, is to “play priest” or “play church”, in the process introducing into the household some sort of ecclesiastical hierarchy (or augmenting the family hierarchy with the ecclesiastical) in an inappropriate way–a way which may ultimately serve to introduce divisions and resentments into your home, and may prove a temptation to see your home devotional time as constituting a separate home church, thus introducing the spirit of schism into your family. Moreover, while your motivation may be pious, the implementation may become little more than vain affectation.
In order to avoid these dangers, my advice would be not to vest in a home prayer service. Moreover, I think it is a powerful teaching opportunity to show your children that their baptismal priesthood doesn’t require robes, vestments and/or regalia in order to be active, potent, and present!
Under the Mercy!
Mark
All is Grace and Mercy! Deo Gratias!