forko

There is no -ork- suffix. Esperanto's affixes are a closed, published list, and the temptation with a word like this is to look for a seam that is not there. Forko is a single root: fork- plus the noun ending -o, and nothing else. The same goes for porko a pig and korko a cork, which rhyme with it and share nothing else. When a word will not come apart, that is an answer and not a failure - see word building on which endings are real morphemes and which are just how the root happens to finish.

It forks the way the English word does. A forko is the thing beside your plate, and it is also the farmer's pitchfork (forkego if you want the size suffix to do the work) and the fork of a road or a river. The verb forki is to fork something; forkiĝi is what the road does when it splits into two - the -iĝ- there is the one that turns an action back on itself, covered in igi and iĝi.

On the table. Forko, kulero and tranĉilo are the set; note that the knife is not a root at all but tranĉi (to cut) plus -il-, the tool suffix, so it is literally a cutting-tool. Manĝilaro is the cutlery as a whole, and if you want the fully homemade word for a fork you can say pikilo, a pricking-tool - but nobody does, because forko is international and short.