"stumble to a halt" is okay. "stumble to a stop" is not okay because of the st...st... It just doesn't sound nice. Stumble is a rather specific verb though. It is used for losing your footing (people or animals' legs giving way). Stumble usually includes some of the meaning of coming to a stop so that it's worth mentioning if you continue after stumbling. "stumbled and fell" is the usual phrase.

You can search for the phrases ("stumble to a halt" / "stumble to a stop", with quotation marks) in a general online search, or even in news articles online. A quick check on Google News tells us that both phrases are acceptable.