Are there addons that provide a timer which starts as soon as you move so you know how long your run has been, and if so, is it allowed if it doesn't tamper with any other game features?

no, not allowed, sowwy's

That's okay. How do I get a timer set up btw? I want to stream it possibly and just want a way for those watching to know the time.

We looked into timer addons a while ago and found nothing conclusive.
Addons and other mods have been banned since.
Luckily you don't really have to worry about timing since the mods time the runs anyway, all you have to do is give a rough estimate.

As for streaming, most people on windows use LiveSplit[1].
Do note that the time LiveSplit gives is not the run's "actual" time,
and mods will retime it anyway.

[1]: https://livesplit.org/

Thank you.