There are a few different ways you can get different mascots. You can receive them from other users in a direct message or in your league chat, or you can purchase them from the Shop.
Accessing the Shop
You can get to the Shop by going to your matchup tab and pressing the icon in the top left corner:
You can also press on your mascot to open up all of your owned mascots, and an option for the Shop will appear in the bottom right.
If your league is in pre-draft mode, you won't be able to view your matchup screen just yet. Don't worry, you can still get to the shop. Under your list of leagues at the bottom, you'll see your profile details.
Press on the cookie amount to get into the mascot shop.
Now, you've entered the shop. Before you buy any mascots, you'll need to have some cookies in your account. You can learn about cookies and how to buy & earn them here.
When you open the shop, you'll see a couple of featured mascots at the top, and then you will see a daily lineup like this:
These mascots are randomized each day in the daily lineup, or you can spend 5 cookies to refresh it now.
You can purchase any of the mascots that are shown in the featured or daily rotation section.
Once you purchase a mascot, you can use them forever in any or all of your leagues.
As you can see in the image above, the mascots have a gift icon to the right of them. You can press on the mascot and it will give you an option to send it or purchase it.
Once you press on "send gift", you can send it directly to a friend or another Sleeper user by searching their name or scrolling through your list. Once you select a user, it will send a DM with the mascot to them.
For sending it to a league, the mascot will be sent via your league chat, and the first person to claim it, gets it.
Gifting Already Owned Mascots
If you want to gift a mascot that you already own, you can only do so once your drafting is complete.
Go to your matchup screen and tap on your mascot as if you were going to change them. You'll see your mascots appear along with a gift icon:
Once you select the gift icon, it will allow you to search for users or leagues, and you can send it that way.