Is there no word for "to exist" in such languages?
I cannot give an answer for all the languages that have no verb "to be". Probably many have the verb "to exist", although it is not this verb that was used by Shakespeare.
As regards Tagalog, existence and possession involve the same two tool-words: may
[mei] for the affirmation and walâ
[wa'la?] for the negation.
May buwáya sa ílog na itó. = There are crocodiles in this river.
Walâng buwáya sa ílog na itó. = There are no crocodiles in this river.
May buwáya akó. = I have got a crocodile.
Walâ akóng buwáya. = I have no crocodile.
couldn't possibly be used to translate what we are dealing with here.
There is magíng
"to become", actually a prefix treated as an independent word. It cannot be used alone; it needs to be associated with a noun phrase. If I used it to translate Shakespeare's line, I'd have to add hárì
['ha:ti?] "king", hence:
Magíging hárì ó hindî ... = To become a king or not ...
The problem is that this sort of translation restricts the portent of the line, etc.