Hi & welcome to TSF.
Whilst I am not completely familiar with your Vanguard engine - many B&S electric starters do not have a "throw in" solenoid - but rely on a "Bendix" to spin the starter gear into the engine ring gear. So for now I am assuming that yours has the Bendix engaging mechanism - as they can "stick" and may be no more than a clean and oil for repair.
1st thing to do is determine if your problem is electrical (dodgy connection) or mechanical -(some part of the starter gear being stuck). When you try to start do you get any noise at all - does the starter motor spin but not engage - or does it do nothing.
If it spins - but fails to engage - it sounds like the Bendix is stuck. If it does nothing - I would be looking for a dodgy electrical connection.
Assuming that it spins but wont engage - then its best to remove the cowl (so you can work a bit easier) then take the starter motor off. At the end of the starter motor - you should see a course threaded spline with a small gear on the end that should slide freely up and down the starter motor shaft (there will likely be a spring in the mechanism to pull the gear back when not in use)
You need to make sure that the shaft, spline mechanism is free to travel and clean and oiled. If that was your problem - this should get you out of trouble. If the spline / Bendix parts are badly worn - you should replace them - but very often - a fix is no more than a good clean and oil.
A word of caution on the electrical connections - make sure you mark where all the wires for the starter motor go - so if there are more than 1 - you put them back exactly where they were.
Let us know how you get on