A wet oil sump is within the engine crankcase. However, how is the oil distributed throughout the engine and all the other magical parts? I know for a dry oil sump, it's with a oil pump which holds the oil from a reservoir, but for wet oil sump, how?
The short answer is "The same way as in a dry sump system." – There is an oil pump which sucks oil out of the oil pan and forces it through the engine oil galleries under pressure.
Wet and dry sump oil systems only really differ in where they store "extra" oil.
Wet Sump Systems
In a wet sump system the oil is stored in the oil pan under the engine. The oil pan serves as both the oil tank (holding the supply of lubricating oil) and the oil sump (where oil eventually collects after making its way through the engine). The pressure pump sucks oil directly from the oil pan and circulates it through the engine's oil galleries.
Dry Sump Systems
Dry sump systems store their oil in a separate tank, which feeds the pressure pump. Dry sump engines have a smaller oil pan which only serves as a sump, and a "scavenger pump" that drains the sump back into the main oil tank. They are termed "dry sump" systems because under normal operating conditions very little oil remains in the sump.
Note that there are other engineering differences in dry-sump systems, but they're largely beyond the scope of your question.
Illustrations thieved from https://www.slideshare.net/jesscar/piston-engines-oil.