Just a small note, but all the references mentioned above are for leagues that have specific rules about Tardy Bowlers. The official USBC rules do not allow bowlers to catch up frames. They instead get 1/10th of the league absentee score per frame up to the frame they are about to bowl, and they can start bowling even into the 10th frame.
Making up frames is for leagues that have specific rules that override the USBC Tardy bowler rules.
109c. Tardy Players
Unless otherwise provided by league rule, a player who arrives late may be permitted to bowl after a game has started under these conditions:
1. The player shall begin play, with the score to count, beginning with the frame then being bowled by the team.
2. The player shall receive one-tenth of the absentee score for each frame not bowled.
3. Partial games shall not be used in determining a bowler’s average, unless league rules require the
secretary to maintain averages based on the actual frames bowled by each player.
45 USBC

Commonly Asked Question – Rule 109c.
109c/1 If a legal lineup is not present by the end of the first frame but a tardy player arrives before the designated tardy player frame, can the late bowler make up missed frames to avoid a team forfeit?
No, a league rule setting a specific frame for tardy players is separate from the legal lineup rule. First, a team must have enough eligible players present for a legal lineup. Tardy players could enter the game if they arrived before completion of the designated tardy frame, but the game would still be forfeited. However, games two and three would not be forfeited since the team would have a legal lineup present.