Is it possible to move a two foot high pawpaw? I know they have long
taprooots. When and how could such a move be done?

There are container grown pawpaws which are that big (or bigger), so
I'm not sure it is hopeless.

I don't really know. Different pawpaw growers seem to have different
opinions about how best to sell them (with the container-grown and the
"dig it up to move" being the two schools). My own approach was to
buy the smallest size our vendor was selling, on the theory that a
large one wouldn't do as well as with non-taprooted trees. (One
vendor of the non-container school is at
http://www.petersonpawpaws.com/ and he says he ships in April, in West
Virginia).

Our pawpaws seem to be doing OK for the first year. But they haven't
put on much growth yet. About the size of our green pepper and hot
pepper plants (in fact, this year the peppers are planted in the same
bed as the pawpaws. The peppers will go somewhere else in the future.
The pawpaws ain't moving).