Ordinarily yes, or at least commit irrevocably, because the application must show funds already at risk. Plan who runs the centre between closing and the visa decision.
Commitment first, then the application
The classification is built around money that has already been placed beyond recovery, which is difficult to demonstrate while a purchase is still conditional. Most files therefore close first and apply second, accepting a period during which the buyers own a business they cannot yet operate in person. Fill that gap deliberately: retain the departing franchisee for a transition period, keep the existing manager with written authority, or agree a short management arrangement, and record it in the purchase documents rather than improvising. The couple may visit for meetings, training required by the franchisor where it does not amount to productive work, and review of the accounts. Operating the shop themselves before the visa is issued is not an option, and doing it quietly is the kind of fact that surfaces at renewal. E-2 status is temporary and provides no direct path to permanent residence.
Hypothetical example: a specialty-coffee roaster has a lease, equipment order and opening date, but the applicant has not yet transferred the final equipment payment. A first review produces a transaction calendar that distinguishes genuine commercial deadlines from optional steps. The lease, purchase contracts and wire confirmations settle what is committed and what remains reversible. If the seller will not accept a risk-bearing escrow arrangement, the buyer must decide whether to close with an authorized interim operator or choose a different opportunity. Starting production personally before status is issued is not a safe way to protect the opening date.