That is a scheduling decision, not an entitlement. An eligible spouse and unmarried children under 21 may seek to accompany or follow, but each person has their own process and eligibility, and no shared date is guaranteed. If the handover slips, an early family arrival can mean months of rent and schooling costs before revenue starts.
Time the family's arrival to the handover, not the signature
Spouse here means a legally married spouse. A qualifying E spouse is generally employment-authorized incident to valid E status; check the admission record, the E-2S notation where applicable, and the proof an employer or agency will accept. Dependent children receive no work authorization from their status.
Residing in the United States or paying household expenses authorizes nothing by itself. Decide who arrives at closing and who waits until the premises are running. Set the family's dates against the enterprise's rather than against the purchase agreement's.
School enrolment deadlines, a residential lease and a spouse's own application each run on separate calendars, and whichever cannot be moved should anchor the others. Where a spouse intends to work, confirm what documentation an employer will accept and when it can realistically be produced, since authorization incident to status still has to be evidenced to a payroll department. Decide in advance what happens if handover slips by a quarter: which household commitments can be deferred, which cannot, and who stays behind.