What's the role of initial endowments in an edgeworth box? I am trying to solve a problem but I don't see the role of initial endowments.
For instance, the question is about finding competitive equilibria.
But there is information about initial endowment.
How should I make use of it?
It's crucial to understand that only initial endowments and preferences matter. Everything else (utility functions, prices, equilibria, core etc) follows from them.

Your initial endowment shows how strong is your negotiation position. 

Situation I. Imagine that the wealth is equally distributed (50/50) and the 1-st player slightly prefers good A to good B and the 2-nd player slightly (to the same extent) prefers good B to good A  than something like this happens.

i.e. you get extra 10 units of A and give 10 units of B (1 to 1 exchange).

Situation II. Now, consider that by the same initial endowment, the 2-nd player prefers B to A much more than you prefer A to B. Then, something like this happens: 

i.e. by the same initial endowment and preferences you get more, now e.g. 35 units of extra giving away 20 units of B (better conditions of exchange as before).

Situation III. Now, imagine that you have only 10% of wealth. Then your negotiation position is weak and by the preferences as in situation I you get much less after exchange. 

p.s. Calculations and plotting were made with pyEdgeworthBox package (I wrote it).