I've been consistently having the same issue, and haven't solved it, but had some ideas for workarounds.
(Though I think its just an inherent bug in inquisit - it would be great if that could be fixed!
1) Easiest solution: I avoid using textboxes if at all possible, and instead will put each question that requires a textbox as an openended trial element on its own page.
Then I turn off mouse input (only keyboard allowed), and tell people to press enter (or control enter, if multiline responses are allowed).
As far as I know you can't do this for survey pages though
/ inputdevice = keyboard
/ numlines = 1
/ linelength = 20
/ buttonlabel = "Press ENTER to continue."
2) If you must use a textbox/surveypage, make sure the textbox is not the first question they have to respond to (captions don't count).
As others have pointed out, on survey pages, after you type in a textbox, the mouse will only appear after you move it over something clickable (e.g. the same or another textbox, the continue button, another question's response such as an option for a radiobuttons element or dropdown list).
When the textbox is not the first question, you actually have to click on the textbox to get the cursor there (vs it just appearing there by default if its the first question). This means that after you type, your mouse will still likely be over the textbox, so it will reappear when you move your mouse to try answer the next question.
3) This one is pretty silly, but may help.
I had one textbox at the top of the screen and the continue button at the bottom, with a large blank space in the middle, so randomly moving the mouse made it hard to move over something clickable.
To increase the chance of randomly moving the mouse over a clickable object, I created a second checkbox question where the response filled much of the blank space, and then made it invisible to people (the text is transparent, and the position of the actual checkbox is off the left side of the screen)
Issues with this are that if your fake question's response overlaps at all with your real questions' responses or the continue button, it will cause issues for making those responses.
If you know your screen size you can move stuff around until it doesn't overlap, but I'm not sure if it will cause issues for online studies where people may have different screen sizes etc. (You may be able to control it through the defaults for screen size and aspect ratio, etc, but I haven't played around with it).
/ txcolor = transparent
/ position = (-30%,40%)
/ fontstyle = ("Arial", 10%, false, false, false, false, 5)
/ required = false
4) For cases where the mouse disappears at the start of the task and you can't click anything, I think with inquisit web links, this may happen if people click on something in the background while the task is loading. Not super sure about this one though.