The crown's blessing might help with that. It is a huge risk though, and if he fails to find an army, Sansa may be in a very bad spot. Given his obsession with her as a Catelyn surrogate, though, he wouldn't have gone down this road without some insurance policy. Maybe the Waynwood debts that got him Harry the Heir in…

Regarding Littlefinger's plan - Who said anything about Tommen? Here's the plan.
1) Littlefinger tells Cersei about Ramsay marrying Sansa, ensuring she doesn't get any ideas about coming to the Boltons' rescue.
2) Roose Bolton brings all of his forces to Winterfell to fight Stannis, probably leaving only a skeleton crew…