How can I unlock museum donations as a non-very first relocator?

Solution 1:

Not sure what the downvotes are about; Game Revolution has an article entitled Animal Crossing: New Horizons’ multiplayer has some major downsides which backs up what you said:

There are some major downsides to Animal Crossing: New Horizons multiplayer gameplay, with players complaining that Player 2 is unable to actually “share” the island in the way they were hoping to. As a result, those sharing an island with another player — the only way to experience Animal Crossing: New Horizons with two or more players using the same Switch — could be left very disappointed.

The article goes on to say:

The first person to create the island is more or less considered “in charge.” That means that Player 2 (or anyone else who isn’t the main player) has some major restrictions for Animal Crossing: New Horizons multiplayer. Other players can’t build buildings or really do anything that advances the story, instead tagging along on Player 1’s adventure. Additionally, the game’s resources aren’t duplicated if another player chooses to play separately on the same Switch. This means that if one player has chopped a bunch of wood or picked up some fruit in the morning, a player hopping on in the afternoon will be left without any resources to gather.

The museum seems to be something that does advance the story and at the moment if you are non-very first relocator it seems like it is off-limits to you.

Also: on day 2 you can start to buy the flimsy axe recipe from Timmy; just work co-op for museum donations for now.

Solution 2:

Secondary players will need to wait until the general store (Nook's Cranny) is ready.

This is Timmy/Tommy's first upgrade, requiring the main player to use the axe/shovel to get 30 stacks of different kinds of wood/ore.

In the store, you can purchase either the flimsy tools outright (for bells) or their DIY recipes.

(I tested this by loading a secondary user on my island, which saw the same selection as shown above.)