I think I may have an STD and want to be checked, can I still get a pelvic exam while menstruating?

Yes and no. Many women hesitate to have pelvic exams during their periods, but their docs generally don't care. The main STD tests (e.g. gonorrhea and chlamydia) and exam of sores are fine any time. However, infections that depend on tests of vaginal fluid (trichomonas, yeast, bacterial vaginosis) are best at other times. If your symptoms are severe, disregard your period and get seen ASAP. Otherwise wait.
YES. Yes, and exam can be done while menstruating, and an investigation for stds can be done. The exam may be limited in some ways, as can testing, but an accurate exam can be done at any time. Some stds can be tested for with urine and some by blood testing as well.