I haven't used it from there but Prince Sakura uses Yamato for their airport delivery service, so I imagine they probably use that for general delivery as well. It's not really they drop it off, the delivery service picks it up from the Prince Sakura and delivers it direct to the Marriott Osaka. The bell desk will take care of the details.

If you have two or more bags, send one bag a day in advance of your checkout, and send the remainder at checkout. Then you'll have your first bag(s) waiting for you at check-in.

You can show them this (but I'm sure just telling then Marriott Osaka will suffice and they will look up the address for you)

大阪マリオット都ホテル
〒545-0052
大阪市阿倍野区阿倍野筋1-1-43
06-6628-6111

Osaka Marriott Miyako Hotel
1-1-43 Abenosuji, Abeno-ku
Osaka 545-0052
06-6628-6111

I like to write in my reservation number in the description box of the delivery slip.