I've heard that George Washington mandated his troops get vaccinated for smallpox, but this must have been well before the germ theory of disease was invented. How did people think that vaccines worked and how were they administered? Was technology advanced enough to make needles?

I hope this will be rigorous enough to stay up, apologies to the mods if my efforts fall short,.

Firstly, there is something of a misunderstanding in the statement that has been circulated regarding the smallpox "vaccination" that Washington ordered the Continental Army to receive: it wasn't a vaccination, but Washington did order an "inoculation" of his troops against smallpox via a method called "variolation".

So we did have a very crude understanding of vaccination at the time called inoculation, and in 1796 Edward Jenner would develop a smallpox vaccine he derived by studying cowpox and realizing if you got that, smallpox didnt affect you. We did have needles to inject substances back then (first recorded intravenous injection was, according to Barsoum & Kleeman (2002), in 1662 in germany) but this was a messy affair that entailed actually cutting a vein open first so sometimes the patient just... died. Bled out or the vein was destroyed. It wouldnt really be until the 1800's that subcutaneous injections happened after a lot of work developing syringes during the course of medical study on cadavers. So no, you couldn't get a shot during the Revolution, it would take about another 75 years for that to be possible. This is all very unlucky for Washington because now, in 1775, the only way to stave off the chance of death by smallpox was to... well, get it and survive. A fair number of the troops in the Continental Army had already had smallpox and lived, like Washington himself who still bore the scars of the near fatal case he battled before he was 19 years old (had a brain-fart and messed up this deatil, thank you to commentor below for catching it). However, not enough men had caught it, and they couldn't let them catch it unexpectedly such that it spread through the army at a inopportune time or inffected a civilian population as the army marched through a town, so the army HAD to have immune troops. Thus, from many possible methods, Washington choose variolation.

So, in order to "variolate" someone against smallpox, you do the following:

Step 1: get a person with smallpox.

Step 2: cut a gash on the person without smallpox.

Step 3: cut a puss-filled boil on the patient with smallpox, dip a bit of thread in the puss, then put that thread into the cut on the OTHER person.

Step 4: hope that the person without smallpox now contracts smallpox, but a much less severe case of it (i can't tell you why, but apparently this method of acquirement made the smallpox case less severe or deadly)

Step 5: quarantine that person for a month while they recover from the (hopefully) milder case of smallpox you caught from the puss-string.

Congradulations! You now have smallpox immunity, and only a few of you died from that "milder" case!

You could also just rub the puss-caked wounds of a smallpox infected person on the wound of a healthy person. Basically, this method was just a controlled way of gaining naturally acquired immunity to the disease so they didnt have to concern themselves with getting infected later. This method was already employed across Europe, and so most of the british army was immune already. Thus when an american soldier signed up, they would be inoculated, recover from their disease and be ready to march out by the time their uniform and equipment came.

Check out Erwin Ackerknechts "A Short History of Medicine" and Elizabeth Fenns "Pox Americana: the Great Smallpox Epidemic of 1775-82" for more reading on the subject.