Can I answer someone's request with: "That, I can do."

That, I can do is not nearly as common as I can do that. Google Books Ngram Viewer shows how much less common it is, at least in printed form.

Having said that, it is grammatical, and it might be used in the context of certain dialogue. You also don't need to remove the comma (as was mentioned in a comment under the question). In the right kind of dialogue, a pause might be natural and a comma warranted:

"I can't do that."
"What about this?"
"That, I can do."

Speaking personally, I'd be more likely to use that expression in that particular exchange than I would I can do that. The reason is that the emphasis is more obviously placed on that if it's the first word in the sentence.

However, as a general statement, and in more contexts than not, the simple I can do that would probably be more common.