Animal Crossing New Horizons – How to Get a Shovel

You should shoot them down with a slingshot (Fandom):

Presents can also appear attached to balloons and must (...) shot with a slingshot.

Do note though, as further down it states:

If shot down over water or flowers, the present will be destroyed and the player will become shocked by it. The present can also be destroyed if it lands in the center of a 3x3 pattern layout.

Fandom also has a nice tutorial/strategy in case you do not have a slingshot:

If the player does not have a slingshot, there's still a way to knock the presents down from balloons. First the player should find a balloon. If it's drifting towards a cliff (depending on the layout of the town, there will be a cliff to the east or the west) he/she is in luck. Wait for the balloon to reach the cliff.

As it passes the cliff, the balloon will lower. The player should get within range of it with a net or shovel out, and use the net or shovel as if he/she was hitting a villager. The balloon should pop, and the present will drop to the beach below. If the player misses, he/she will have a moment or two to try again.

This strategy will not always work, as the balloon may be heading in the direction away from the cliff. If so, the player will have to wait and try again with a balloon heading the other way.

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.