Les did it. Chet did it. Hank (Marvin) did it more, but nowhere near as much as Jimi at Woodstock!!!!!
They used the whammy quite creatively, and look ma, no string locks!, without excessive tuning problems.
So, why were string locks invented?????
Because, players didn't know how to string their axes properly, and more importantly, either having the knowledge or patience to streeeeeetch their strings when they changed the buggers!

My method is to curl my fingers under the string, and using the index finger and thumb, pull the string about a finger width above the fretboard, and run up and down the length of the string. (Otherwise only a section of the string will be stretched!)
Retune, and restretch til the tuning becomes stable.
Next, change the next string, and repeat the stretching /tuning procedure. Retune all the strings, not just the one you are stretching. Especially on a strat through-body bridge with whammy configuration, string tension and stability is dependent on the whole guitar being in tune.
When all six strings are done, check tuning, and obviously, restretch and retune where required.

Always drop the pitch, and tune up to the note, never just tune down.

Sometimes strings can bind in the nut, especially when the nut is plastic, and strings may tune themselves sharp.
Plastic needs some humidity, and our guitars are often in artificially heated areas with very dry air.

I keep a plastic squeeze bottle of powdered graphite, (which is what pencil leads are made of) to insert in the nut slots if I have string binding problems. Locksmiths and hardware shops should be able to sell this to you, as the powdered graphite is used for squeaking door hinges.

After this, happy whammying, even without string locks!