There is no question type to do that out of the box, at least not with validation.
Here a few options you can use, but might need custom validation:
Array - text: respondents would write their time freely. Either you restrict the input via regex, but you need to explain the format very detailed to them, which is not ideal, or you let them write the time as free text and then need to check on the format and "normalize" it when analyzing (quite a bit of work probably).
Array - numbers with dropdown mode: If you don't need every minute of the day but go in hourly, half hour or 15min steps you could use an array number question with dropdown mode and give them the options: 09:00am, 10:00am, ..., 11:00pm, etc.
However, this question has nothing to do with "installation and update issues", which is why I will move it to the "can I do this with Limesurvey" forum.
Help us to help you!
Provide your LS version and where it is installed (own server, uni/employer, SaaS hosting, etc.).
Always provide a LSS file (not LSQ or LSG).
Note: I answer at this forum in my spare time, I'm not a LimeSurvey GmbH employee.
Not available out of box with LimeSurvey. Perhaps someone knows a workaround. I'm not aware of one right now.
What version/build of LimeSurvey is used?