Playing tabletop role-play games without dice is something that happens. Dungeons and Dragons is apparently quite popular in prisons where escapism is valuable for mental well-being and dice (otherwise used for gambling) are prohibited. I've played games during car trips where rolling a die isn't possible or practical, so the adventures have been more focused on narrative than combat. I've also seen games where people write the numbers 1 through to 20 on paper and just pull them from a bag, or games where a pack of cards is used. I have a number guessing method, which I've never seen mentioned elsewhere, but assume others could or have come up with something similar.
This guessing method replaces the most common die in D&D, the twenty sided die (known as the D20). Normally, the person running the game (the Dungeon Master, or DM) would set a difficulty between 1 and 20 for a task the player is trying to perform. A 10 might be considered an easy task, where a 20 would be remarkably difficult. The player would roll the D20 and add any modifiers to their roll based on their character's abilities. A character who is particularly athletic might add 5 to any attempt to climb a wall for example. If the roll is equal to or higher than the difficulty value, the player succeeds.
Without dice, the DM sets the difficulty of the task just like before, but now subtracts that number from 21 giving a new value we will call the Difficulty Range. The smaller the Difficulty Range, the more challenging the task. The DM then secretly chooses a “random” number between 1 and 20 and adds the Difficulty Range value, retaining the original number. If the new number exceeds 20, assume that the value has gone “around-the-horn” and continue counting from 1. These two values give the band the player will attempt to guess in order to be successful in their task. We will call this band the Target Band. Lastly, the player openly picks a number between 1 and 20, creating their own range by adding their character's ability modifier. We will call this the Player's Range. If the Player's Range intersects the Target Band, they are successful.
This system does present both DM and player with a little bit more mental gymnastics compared to simply rolling dice, but when no dice are available, the game may continue.