This is one of those situations where English doesn't necessarily make sense.
This sentence is perfectly normal:
✔ He walked with a limp.
This would imply that limp is countable. But, syntactically speaking, the word itself isn't. (We simply never pluralize the noun directly.)
So, we would never say the following:
✘ He walked with two limps.
Instead, we would use the following:
✔ He walked and limped twice.
✔ He walked in a series of two limps.
In the first version, it's the verb limp that's being used. In the second version (it's syntactically correct—although a little odd to refer to a series that consists of just two occurrences of something), the noun limp is pluralized but only because it's being modified by an adjectival phrase that comes before it.
Note also that because it's not directly countable, this is also wrong:
✘ He walked with one limp.
This means that this is one of those somewhat uncommon cases where we use the indefinite article in front of a mass noun. It's also the case that this particular mass noun can't take the definite article:
✘ He walked with the limp.
So, we would not use your version:
✘ He walked with limps.
However, we could express it differently:
✔ He walked and limped.
✔ He walked while limping
✔ He walked in a series of limps.
Note that, this time, using series isn't awkward because the implication is that there were a lot more than just two instances of limping.
One other reason why he walked with limps sounds strange is not because of us not pluralizing the noun directly, but because it almost sounds as if limps is the name of a person—making the interpretation (which, in context, is obviously wrong) he walked alongside Limps.
If you want to emphasize a problem with each of his legs, rather than a particular number of instances of limping, you could say:
✔ He walked with a limp in each leg.
✔ He walked and limped on both legs.