Can I impregnate MDF with something to make it dishwasher proof?

If you are using this cutting board for cutting food, STAY AWAY from
MDF. MDF is made up of sawdust and a glue which contains
FORMALDEHYDE so you certainly don't want food to come in contact.
You also mentioned a dishwasher ... wood in a dishwasher will dry
out and be damaged. If you are buying a plastic sheet material as
someone else discussed, do your homework as some plastics which have
an oily feel to them may also have toxic chemicals. Personally, I
would stick to a kiln dried pine. It's soft so not hard on your
knives.

Go
od luck