Well "Eisiau" actually isn't exactly a verb but a noun which we happen to use to express "To want". Though to answer your question, unlike in English, a form of "To be" is needed in this sort of sentence. It more accurately translates as "The dog is wanting to eat an apple" though this construction is not used in English so instead we translate it into "The dog wants an apple".
A better way to start to understand this pattern is to add a missing element:
- *Mae'r ci ag eisiau bwyta afal" - literally 'The dog is with (has) a want of eating an apple'
The ag is not seen except occasionally in a more formal register of the language.
In the informal register of the language there is no distinction between the present and present continuous. In the formal registers the tense system is different in some respects.