I agree with you: "the process how to […]" sounds wrong to me.
This seems like the sort of thing that might vary between speakers, so as a sanity-check, I've looked through the first few dozen Google-hits for "the process how to" [link], and found that only one of them was actually using the construction that you're interested in. (In the rest, "the process" and "how to" belonged to separate clauses.) That one example [link] is by a non-native speaker (a sushi restaurant in Indonesia that also writes of "An experience of choosing fresh first pick seafood") [link], which supports the notion that this is just a non-native-speaker error.
That said, I'm not a huge fan of your proposed fix ("the process of how to […]"); it's definitely grammatical, but it's a bit redundant to have both "process" and "how". Also, I think I prefer "process for" over "process of" when describing bureaucratic procedures for accomplishing a goal. So, I'd suggest either of the following:
"Do you know how to get a temporary emergency passport?"
"Do you know the process for getting a temporary emergency passport?"