In quadrilateral $abcd$, we have $ab=3,$ $bc=6,$ $cd=4,$ and $da=4$. if the length of diagonal $ac$ is an integer, what are all the possible values for $ac$?

The triangle inequality applies.

In order for ACD to be a triangle, the length of AC must lie between CD-DA=0 and CD+DA=8.

In order for ABD to be a triangle, the length of AC must lie between BC-AB=3 and BC+AB=9.

The values common to both these restrictions are numbers between 3 and 8. Assuming we don't want the diagonal to be coincident with any sides, its integer length will be one of ...
{4, 5, 6, 7}