I'm not a lawyer, but I don't think that he can do that.
Bill Clinton had a legal defense fund, and if memory serves, it paid not just his legal fees but his fines. DeLay had one. I don't think the government has a lot to say about any donations folks want to make to a worthy person, group or cause, though they can decide whether the donations are tax-deductible or not, and they can shut down "charities" that launder drug or Mob money or fund terrorism.
It's a complicated question. On the one hand, Thompson has a legal obligation that could prevent him from running for president. NBC has a legitimate case; if Thompson violates his contract and thus costs them millions, they're owed compensation. If a group of donors wants to remove that impediment, because they think he should be president, why not?
On the other hand, such a fund would be a way folks could do a very large favor for a presidential candidate, possibly expecting large factors in return, outside the limits and disclosure requirements of campaign finance. Both Clinton's and Bush's presidential library funds have been eyeballed as possible means of, not to put too fine a point on it, laundering a bribe.