Yes, they can. In fact, one of the first theories of cancer was that it was caused by viruses, although now we realise that there are other causes of cancer. There are quite a number of viruses, called retroviruses, which work by inserting themselves into human DNA. They can disrupt the normal regulation of cells, and cause cancer. In fact, the recent ‘cervical cancer vaccine’ isn’t really a vaccine against cancer. It is a vaccine against a virus that causes cancer.
Viruses can certainly cause cancer: for example, HPV (human papillomavirus) can cause cervical cancer (so when you get the cervical cancer jab, you’re actually being vaccinated against this virus) and hepatitis viruses can cause liver cancer. But the vast majority of viruses do not cause cancer, and as far as I know, no bacteria cause cancer.
There are a series of viruses that are called transforming viruses, for their capacity to transform a cell into a cancer cell. This means that these viruses are able to induce in different ways those mutations that lead to the cell starting dividing indefinitely. One is the papilloma virus (HPV), whose infection greatly increases the risk of developing cervical cancer or the Epstein-Barr virus (EBV) that is strongly linked to Burkitt’s Lymphoma