Is the following correct?
Both of these essentially act as a WebKit wrapper.
Or should it be the following?
Both of these essentially acts as a WebKit wrapper.
The word both is a determinative (in the terminology of CGEL). Determinatives are not per se singular or plural (except these and those), but they select singular or plural nouns. When both appears without the head noun, as it does here, there is a fusion of functions and it becomes both the determiner and the head of the plural NP. The verb then agrees with the plural NP.
(ignore the clumsy grey line above Head)
It acts only like plural because, unlike each1 and none2, "both" refers to all of the elements and not to one of them.
Read these examples from the OALD:
Note: for more accurate explanations, visit the links, otherwise if I add too much info, this answer gets off topic.