At first, I thought "me rogo," but the dictionary did not confirm my suggestion. I think my German is interfering ("ich frage mich").
In my opinion “I wonder” is such a versatile expression in English that it is futile to wish for a direct Latin equivalent that would cover all uses.
For example, with sentence questions, “I wonder” often means little more than “maybe.”
I haven't seen the cat in a while. I wonder if he is sleeping.
Felem aliquamdiu non vidi. Nescio an dormiat.
On the other hand, it can also mean “I'd like to know.”
The cat is nowhere to be found. I wonder where he is.
Feles nusquam reperiri potest. Cupio, ubi sit, scire.
In short, avoid translating single words or expressions, and try to capture the overall tone and meaning.
Based on German "Ich frage mich", I started out with inquiro, but that does not cover it. Then I wondered: would miror do?