Can my 15-month-old get a rash and fever from vaccines?

Yes. Yes - vaccines cause an immune response to what is contained in the vaccine. This may, and may not, include having a fever. Usually, a fever related to vaccines will last up to 3-4 days and not be terribly severe. Any fever that is very high, or any time the child is acting very ill, even after vaccines, you should discuss the situation with your doctor. The reaction may also include a rash.
Yes. A small number of children get a rash and fever 7 to 14 days after the measles-mumps-rubella (MMR) or the varicella vaccines. The rash is usually on the main body area and lasts 2 to 3 days. Call your healthcare provider within 24 hours if the rash lasts more than 3 days or gets itchy. Call your child's provider immediately if the rash changes to purple spots.