Ooh a question I can answer!

Smallpox was a big problem in the American Revolution, but only really for the Continental Army and for the colonists. British soldiers were much more likely to have had smallpox as children, leading to immunity, and if they didn't then they were innoculated with smallpox.

I'd like to make a quick clarification before getting down to brass tacks. Innoculation in this time period can technically refer to being purposefully infected with smallpox OR cowpox, but it often refers only to the former. Smallpox innoculation specifically was called variolation, and cowpox innoculation was called vaccination (from the Latin word for cow). Both led to immunity from smallpox. For the purposes of this post, whenever I bring up innoculation from here on out, I am referring to variolation.

Washington had a choice to make: don't innoculate the Continental Army and risk entire fronts being brutalized by smallpox, or innoculate the whole Continental Army and risk having large numbers of soldiers out of commission for a full month (the normal length of a bout of smallpox was ~24 days). So Washington looked at the pros and cons of each.

Pros:

Avoid a repeat of the campaign into Canada (I believe Quebec specifically). During this particular campaign Continental troops were fighting in a cold and harsh frontier against the British and they were dealing with a large smallpox outbreak at the same time. This campaign saw a lot of death, a lot of suffering, and a lot of desertion.

Avoid a repeat of Boston. During the war, the city of Boston saw a huge outbreak of smallpox leading to it being quarantined and by Continental troops because they couldn't risk an outbreak among the soldiers. Only the immune could enter and exit. I might also mention offhand that smallpox in North America was seeing a big flare about every 20 years and last a while, and this particular outbreak was from 1775-1782.

Avoid Lord Dunmore's folly. Lord Dunmore was of course a British commander, and he offered freedom to slaves and indentured servants in Virginia who would fight for Britain. Unlike British regulars, these recruits were not immune to smallpox, and Dunmore chose not to innoculate them. Called the "Ethiopian regiment" they went all around the Chesapeake region in a flotilla. 650 men, all vulnerable to smallpox, put together on boats. It went about as one might expect. The vast majority of them came down with it, all while new recruits were coming in, 8 or 9 a day! Lord Dunmore wrote in June of 1776, "I should have had two thousand blacks; with whom I should have had no doubt of penetrating into the heart of this colony." He started out with 650. At least 1,350 others joined and died, according to Dunmore.

Cons:

Lots of soldiers will be out of commission for month at a time, meaning that if they were to go through with it, they would have to stagger it by unit. If that's done it could weaken whichever front that unit is from.

Not all soldiers in the Continental Army wanted to be innoculated. There was a strong fear of desertion if innoculation were to become mandatory. It did end up happening to a degree, but it wasn't as bad as expected.

Soldiers undergoing the innoculation would be vulnerable to British attack. If something on the front went wrong and the British made it to the encampments full of soldiers undergoing innoculation, they'd be unable to put up any real resistance. Naturally contracting smallpox would completely incapacitate a person, and contracting it by innoculation would still make a person significantly weaker compared to someone healthy. The British would have no problem entering these encampments because the British were immune. So there was a real fear of these innoculated soldiers not succumbing to the disease, but to the enemy.

So what did Washington decide to do? He took these fears and potential benefits into account and decided to innoculate the Continental Army. They staggered it so that at a given moment the large majority of the army was still functioning. Did people desert? Yep, but it was worth it. Did the British attack encampments of sick troops? Not from any source that I've seen. Did it have an effect on the fighting capacity at the front? Not that I'm aware, but maybe.

Source: Elizabeth A. Fenn, Pox Americana: The Great Smallpox Epidemic of 1775-1782.