When we opened applications at the end of June, about 500 seats were reserved in a little under a week.
Literal
June end [target-に] participants [object-を] recruit [conditional-たら] one week little-less-than [with-で] approximately 500 seats [subject-が] reservations [copula-で] full [target-に] became.
An event-planning retrospective. 募る ('to recruit, solicit, invite applications for') is a specific word for gathering participants or contributions. 一週間弱 uses the suffix 弱 ('a bit less than') to indicate approximation; the opposite suffix is 強 ('a bit more than'). 予約でいっぱいになる ('become full with reservations') uses で ('with') to mark what fills. Typical business/event description.