Verb to use with "workload"

Native speakers would most likely use the verb "take on" for a noun like workload, as it is viewed as a type of obligation or chore (as one would "take on" a debt or a dependent). Note that native speakers would also use the indefinite article in your sentence, as a workload is a measurable single thing. So, they would say:

I wish to take on a heavier workload.

Of course, most people would rarely want to express that sentiment, regardless of language.

I would welcome a heavier workload seems, idiomatically, to express what you are wanting to say.

I would say

I wish to be given a heavier workload.

Or

I wish I could have a heavier workload.

"Take on" suggests too strongly (to me) that the extra workload will be pushing me to my limits. Which may not be the case.