Why won't my sims 4 load my gallery

Have you tried going to Game Options > Other to make sure both enable Custom Content and Mods and Script Mods Allowed are checked?

Mods get disabled with each game patch and need to be turned back on each time there is an update.

If that doesn't help, have you checked that your connection isn't being blocked. Some places block connections, like colleges.

If you have a desktop maybe you can take it to a friends/relative's house to check to see if you can connect to the Gallery using their internet.

If you have a laptop, try a local coffee shop, restaurant or library to see if you can connect to and use the gallery.

Whether using a laptop or desktop computer you can use your mobile phone device as a temporary Wi-Fi hotspot for internet connectivity.

If you find you can log into the gallery from some where else, you may want to reset your router and/or modem.

Try repairing the game. - In Origin > Games Library tab: Right-click on Sims 4 and select Repair.

Turn off Origin In-Game:
Open Origin, Select Origin > Application Settings > Origin In Game tab and click the on/off switch to disable Origin In Game. - This deactivates Origin In-Game, as it has been known to cause problems with games.