My main concern is that this will cause me to start off on the wrong foot with the company, and jeopardize my position there since I would be taking the time off before establishing myself there.
It makes sense to be concerned about taking that much time off so early in your job tenure. In some companies you would still be in "onboarding mode". Being out that long is likely something that needs to be planned and mitigated.
Is there anything I should do about this?
You should discuss this with your boss now. They may want to alter training and/or project plans to account for your absence. The sooner they can do this, the less the impact. Bosses don't like surprises.
Would it make sense to ask for input from HR or my new manager on how much trouble this would cause?
Yes. Input from your manager might alleviate your concern or help you make whatever decisions you need to make.
If you need input regarding your insurance coverage or your rights regarding time off, talk with HR.
What should I be taking into account when assessing the potential career impact of this decision?
As with all decisions, get all the information you can before deciding.
Most likely, taking this time off will be fine - certainly in the long term, and likely even in the short term. Still, your manager will appreciate knowing about it as soon as possible so that they can alter their plans as needed.
Talk to them now. Don't wait.