Mutations are caused by the insertion of the wrong bases during replication, or by chemical changes to bases, either by chemical agents or by radiation.
The rate of mutations could be increased by exchanging the E.coli polymerases with other polymerases displaying higher error rates (or with less proofreading capabilities). A lot of errors are quickly repaired via different mechanisms, knocking out all or a few of those mechanisms might result in an E.coli strain with an increased mutation rate (if they survive).
A more elaborate way would be to engineer E.coli to contain a pathway that synthesizes a mutagenic compound. The easiest way to to this would be 1 enzyme that generates reactive oxygen species, as those also cause DNA damage. You could also delete some of the catalases and other enzymes that remove reactive oxygen species instead.