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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?

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First, according to MWG, Def 15.B.1, "a Walrasian (or competitive) equilibrium for an Edgeworth box exonomy is a price vector $p^\star$ and an allocation $x^\star = (x_1^\star, x_2^\star)$ in the Edgeworth box such that for $i=1,2$:"

$$ x_i^\star \succeq_i x_i^\prime \qquad \text{for all} \quad x_i^\prime \in B_i(p^\star) $$

Typically in such problems, you must have the Walrasian demands for each consumer and good, and the initial endowments for each consumer. From every endowment, $\omega_i$, you can infer the individual's wealth, just and obviously multiplying the endowment's quantities for each good with the respective prices. For example, having $\omega_1=(2,3)$, means that the individual 1 is endowed with 2 "units" of good 1 and 3 of good 2. So his/her wealth would be $w_1 = 2 \cdot p_1 + 3 \cdot p_2$.

With these, you can proceed to find the Walrasian equilibrium; however, a more detailed question on your behalf would be more enlightening.

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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.enter image description here

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:enter image description here

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.enter image description here

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

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In the budget constraints of the competitive equilibrium.

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