What size envelope for a 5×7 card?

A7 – 5¼ × 7¼ inches. It gives ⅛″ of room on each side, the standard clearance for announcement envelopes.

Will an A7 mail with one stamp?

By size, yes: 7.25 ÷ 5.25 = 1.38, inside USPS's 1.3–2.5 aspect-ratio range, and both sides are within the 11½″ × 6⅛″ letter maximum. Cards with ribbons, wax seals, or embellishments that make the piece rigid or lumpy become nonmachinable and need extra postage. Heavy invitation suites can go over 1 oz, so weigh one.

Alternatives

• Thick card or multiple inserts: try a A8 (5½ × 8⅛).
• Want a liner or a euro flap? Still A7; liners don't change the outside size.
• Outer + inner envelope (formal invitations): inner A7, outer "A7½" / "Outer A7" (about 5½ × 7½); confirm with the seller.