Is the expression "having a good time" too personal?

"We hope you're having a good time on your vacation" has a casual vibe to it. I wouldn't say overfamiliar.

It depends how well you get on. Are you on first name terms? Do you ever joke with eachother? Have you had a business meeting over a drink or something?

You should really get a feel for which one is the most appropriate based on your relationship with this person.

"We hope you're enjoying your vacation" is what I personally would say to a business partner unless we knew eachother on friendlier terms. It is cordial, but also professional sounding.

Both sentences are equally correct. I would argue that having a good time does sound a touch less formal, but it is in no way informal or inappropriate. Also, I hate to be a pedant, but your first sentence ought to start with A colleague and I since you are performing an action.