Is it alright?
Yes, probably. From a paranoid standpoint, it tips your hand that you're doing something. You could be applying for a grant, grad school, or a new job - so it's not necessarily that you are planning to quit soon. But someone asking for a copy of the transcript is bound to be a bit of a puzzle in the HR department.
Is it efficient?
Probably not. If you think you will get the copy from your work quickly, you are mistaken. The HR department doesn't have a vested interest in getting this for you quickly, and it may well be hard to find. Don't assume that this is the fastest path if you need these documents in a hurry - try multiple paths, including calling your school.
Is it weird?
In some situations, the new position will be looking for a verified transcript that is send directly from your school to the employer. It's generally not assumed that your current employer is the best place to get paperwork that is relevant to your time at a different institution (your school).
In essence, it's not your employer's job to be the keeper of your paperwork. It's your job. When you give a document that you have 1 original copy of, give a copy, not the original. If the next employer insists on an original, then insist that you have the time to request it from your school.
It's not at all unreasonable to say that 1 day of turnaround time is NOT reasonable for paperwork like this. My approach would actually be to go to the interview with a printout or copy of my transcript & diploma - whatever I can find online, and to promise to have already requested a formal dispatch from the school that will arrive in whatever number of days the school can manage.