in the meaning of "working hard" : do we say: grapple with .... grapple on .. OR grapple for ... ?

You grapple WITH.

In that context, the term "grapple with" [insert subject of difficulty here].

Grappling is synonymous with "wrestling", and can be used in a similar fashion. You can grapple, grapple on a surface, or grapple with an individual. You can even grapple for something or someone (like money, or recognition).

Grapple with. As in:

Today, accountants "grapple with" tax increases for their clients.

or...

Sole-traders "grapple with" the hard-labour put into working alone.

The word grapple isn't normally associated with working hard, it usually means to contend with something, usually a fight or a struggle to do something. For instance, I grappled with the robber until he gave in and ran off. or I grappled with the chain on my bike and eventually got it back on. I think you might be better using the word grafting or graft which is usually associated with hard work.