According to Utah State Code, you can attend daycare or public school, Pre-K, through higher education, with a religious, personal belief, or medical exemption. There is an exception for higher education students in medical settings. These students still qualify for religious exemptions, under Title VII of the Civil Rights Act. (Please follow the links in the text above, and below, for more information.)
You do NOT need a doctor’s note for personal or religious exemptions. You can exempt your child or yourself from any or all vaccines. Because the COVID shots are not yet added to the paperwork, I would suggest adding them yourself, and getting the form notarized, before turning it in to the school. (I am not an attorney and this is not legal advice. This is just what I would do, in this situation.)
To get an exemption for Pre-K-12, review the indoctrination module; choose which type of exemption you want (I suggest religious); and print out the form.
This is currently no mechanism for exemptions from higher education vaccine requirements, so vaccines cannot currently be required for those who qualify for standard exemptions.
For more information on Utah vaccine exemption law, please refer to this video. (This will be updated shortly, based on 2021 legislation, but higher education is now covered by vaccine exemptions, with exceptions noted above.)