。
At the early-morning reception, we drew lots to decide the order.
Literal
Early-morning [of-の] reception [at-で], lots [object-を] draw, order [object-を] decided.
クジを引く 'to draw lots' is a fixed collocation — クジ can be written くじ, 籤, or in katakana depending on style, historically referring to strips of paper used both for fortune-telling at shrines and for secular random selection. The te-form 引いて links 'drawing lots' as the means for the following 'deciding the order,' a common purpose-linked sequence without an explicit 'by' preposition.