The bowl is the subject. It can be unambiguously reworded like this:
The bowl was lifted without it spilling its contents.
This sentence is identical in meaning, and it makes clear that the subject of "spill" is the bowl itself.
It would be impossible to have the unnamed person be the subject of "spill". Let's change the sentence a bit to make that clear:
The bowl was carried without tripping.
The first read of this sentence is that the bowl didn't trip, but a bowl cannot trip, so it can only refer to the person lifting the bowl. With that reading, "tripping" becomes a noun referring to the concept of tripping, rather than a verb with a subject, along the lines of this version:
The bowl was carried without any tripping happening.
It cannot be understood this way:
The bowl was carried without him/her tripping.
There is no understood antecedent for "him/her", so it doesn't make sense.