Hi, I'm in the middle of an apprenticeship.
Finding an apprenticeship isn't as hard as some people like to make out. Years finding an apprenticeship? I can only imagine that there's some vibe you'd have to give off to put off so many plumbers.
If you are punctual, polite and well turned out there's no reason someone won't give you a start. Being meticulous is a good attribute as is a sense of humour.
Know when to shut it as well

No-one like a cocky teenager.
Also consider why you want an apprenticeship and be a plumber.
Saying you want to learn from first hand experience instead of paying to take a fast track college course is a good start.
So is because you want to earn an honest living by doing something practical but also technical.
Being able to stand back and derive satisfaction from a job well done is a good reason.
My employer has shown me a couple of letters he's received and they're cringe worthy. One guy couldn't spell apprentice (apurentise) and the other guy was gushy about how he'd like to start his own firm as soon as he's qualified.
If you speak to a plumber and he turns you down on the basis that he's already training someone there's no harm in asking when his existing apprentice's training will be complete and whether he'd consider taking you on afterwards.
I know there are few people looking to expand at this time but you never know.
Offer to forward your details for consideration when times are looking up. Chances are it'll go in the bin because he doesn't have time but you never know. The more you send, the more chance that someone will hold onto it.
Above all don't ever send a text message asking for a job and always ask on the phone if now is a good time to call. Answering a phone whilst on a ladder is no fun
Best of luck!