Yes, you can take radiographs on patients with Hashimoto’s. Just like any patient, be sure to use a thyroid collar when exposing radiographs. Hypothyroidism may lead to delayed wound healing, so radiographs are very important in determining bone levels and assessing periodontal health (along with other assessments: perio chart, etc.) Here is a link to research that may help:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3169868/