This 2-category is $T\text{-Alg}$ for a 2-monad $T$ on the complete 2-category ${\rm Cat}^{\rightrightarrows}$. Thus, by results in section 2 of Blackwell-Kelly-Power's "Two-dimensional monad theory", it admits products, inserters, and equifiers, hence all PIE-limits.
I suspect you could generalize this to a suitable 2-category by Yoneda-embedding it into a presheaf 2-category so that there is a 2-monad to talk about, but I don't know offhand where it might be written down. (Although to define weak internal categories -- at least, of the sort that specialize directly to pseudo double categories -- you also need strict pullbacks, which aren't flexible.)
The generalization to flexible limits would follow from the first few remarks in section 7 of Bird-Kelly-Power-Street's "Flexible limits for 2-categories", which claim that $T\text{-Alg}$ admits flexible limits whenever $T$ is a flexible 2-monad, and that this is the case whenever $T$ has a presentation that avoids equalities of objects. However, these remarks refer to a paper in-preparation called "Flexibility for 2-monads" which as far as I know never appeared. The needed results may exist elsewhere in the 2-category theory literature, but unfortunately I don't know where exactly.
Edit: It looks like the missing results appear in Lack's papers "On the monadicity of finitary monads" and "Homotopy-theoretic aspects of 2-monads".
