Any job that can be done remotely is doable by someone with a mental disability, unless that disability is so severe that they cannot function adequately enough for the job*, or where they'd put someone else at risk of harm (that decision is up to their doctor).
The Ideal Situation
In the UK, it's a requirement of the Equality Act and Disability Act that businesses accommodate for people with disabilities (by way of special equipment, and so on). I've done quite a bit of work with mental healthcare specialists both in regular work and while freelancing for private mental healthcare institutions contracted out by the NHS, and the vast majority of businesses don't actually adequately provide for someone with a mental illness of any description.
Introverts and those who are slower have the potential to benefit extremely well from remote working, as they don't have people pushing them to do stuff right this second, avoiding situations that usually put them into panic mode.
From personal experience, though, unless that person has constant contact with other people, it can end up seriously damaging them. If anyone with a mental illness wants to remote work, seriously consider having a schedule you/they stick to that involves going outside and doing other activities. I have been there before. It's not good.
Provided they have adequate care, remote working shouldn't be an issue. Just make sure they are well-supported in doing so; UpWork and so on are very unforgiving places for those less experienced, and I would strongly recommend remote working for a company as an employee, not a contractor, due to the protection offered by law for disabled persons.
* Yes, I am aware this is dodging the question slightly and is extremely subjective.