Webb needn't bother running as an independent
The idea is that especially with Hillary Clinton and Donald Trump as his opponents Webb's considerable defense and foreign policy chops would attract the support of genuine conservatives uncomfortable with Clinton's record and Trump's manifest ignorance.
I like and even admire Webb in some ways. The trouble is his social policies, which are the standard far-Left Democratic pro-choice, pro-marriage redefinition, pro-social chaos mess. I very much doubt that many conservatives who would find themselves unable to vote for Trump would vote for Webb instead. I certainly wouldn't.
I continue to think that a centrist third party bid would be a good idea, just in case somebody like Trump or even Ted Cruz or Rand Paul- someone clearly not presidential timber- were to get the GOP nomination. But it would have to be somebody with positions on social programs people unable to vote for Clinton and turned off by Trump and Cruz and Paul could get behind.
Anybody who could swallow Webb's social positions will probably vote for Clinton. If he's betting on a substantial number of Democrats who don't march in ideological lockstep with The Agenda, he's going to be sorely disappointed on election night.
The only sizable group of voters in the scenario I've laid out who will feel themselves disenfranchised will be conservatives left without a credible candidate to support because a non-credible candidate like Trump or Cruz or Paul has been nominated. And those folks would find Webb just as unacceptable as the other choices.