Yes.  But the gifts would probably not be deductible.  The organization will probably not qualify for charitable status.  (See Ready Reference Page: “What Do We Mean When We Say ‘Nonprofit’?”)

In most states, a “nonprofit” may be formed for any legal purpose.  In those states, it would be legal to form a nonprofit to pay legal fees (thank goodness for the profession!), but soliciting contributions to pay fees for a suit for or against certain specific people is not likely to be considered “charitable.”  An organization with a broader social welfare purpose might conceivably qualify as exempt under section 501(c)(4) of the Tax Code, but it is unlikely that a group limited to paying legal fees for a specific lawsuit for or against certain people would qualify.  Even if it did, contributions to a (c)(4) are not deductible.

In the current climate, such an organization might qualify as a political organization depending on the parties involved, but contributions to political organizations are not deductible either, even if the process would not violate election law.

Once again, calling an organization “nonprofit” is not definitive.  If you don’t know what box the organization is in, you don’t know what rules apply.