At what stage does a deer tick get lymes?

Lyme disease is caused by a group of spirochete bacteria called Borrelia burgdorferi sensu lato. Ticks can get infected by these pathogens while sucking blood on an infected vertebrate. As ticks (well, not all ticks, but the ones that carry Lyme disease) feed at every stage of their lives on a different host, they can get infected at every stage.

Some pathogens can circulate transovarially (i.e. an infected female tick's eggs are already infected), but that's not the case with Lyme disease.

So basically a larva ('first stage tick') can't be infected until it has fed.

Of course the chance of a tick getting infected while feeding on an infected host is not 100%.

It's important to know, that if a tick once get infected with Lyme disease, it stays infected all it's life.

Plus there's a small chance that a tick can get infected on an uninfected host by 'co-feeding'. That means that a tick basically sucks up the saliva (and also the spirochetes in it) of another tick that's eating nearby.

So, the short answer to your question is: ticks can get Lyme disease at every stage.