Just some quick addenda to David's comments:

1. You can see that the automorphisms of a Lie groupoid do not form a richer (groupoid) structure in some more specialised examples. It is well known that proper etale Lie groupoids are a Lie groupoid formulation of orbifolds (they are often called orbifold groupoids). Now their automorphisms corresponds to the group of orbifold diffeomorphisms and these can be turned into an infinite dimensional Lie group (this was the whole point of my PhD thesis, see The diffeomorphism group of a non-compact orbifold). The point is that these objects really form a group and not some higher structure like a groupoid or a $2$-Lie group.

2. In the literature people have considered groups of smooth maps on the arrow space of a Lie groupoid. Basically you look at diffeomorphisms Diff$(G)$ which preserve source and target fibres. Again this yields groups not groupoids and has the advantage that the bisection group of the Lie groupoid identifies a subgroup of the automorphism group, called the inner automorphisms (by sending a bisection to the associated left translation). To my knowledge no smooth structure has yet been constructed on this automorphism group in general (though it seems to be an interesting question if there is one). For more references let me sneak in my recent paper The Lie group of vertical bisections of a regular Lie groupoid. The references can be found in the introduction together with an explanation of how this is related to the automorphism group.

Just some quick addenda to David's comments:

1. You can see that the automorphisms of a Lie groupoid do not form a richer (groupoid) structure in some more specialised examples. It is well known that proper etale Lie groupoids are a Lie groupoid formulation of orbifolds (they are often called orbifold groupoids). Now their automorphisms corresponds to the group of orbifold diffeomorphisms and these can be turned into an infinite dimensional Lie group (this was the whole point of my PhD thesis, see The diffeomorphism group of a non-compact orbifold). The point is that these objects really form a group and not some higher structure like a groupoid or a $2$-Lie group.

2. In the literature people have considered groups of smooth maps on the arrow space of a Lie groupoid. Basically you look at diffeomorphisms $\operatorname{Diff}(G)$ which preserve source and target fibres. Again this yields groups not groupoids and has the advantage that the bisection group of the Lie groupoid identifies a subgroup of the automorphism group, called the inner automorphisms (by sending a bisection to the associated left translation). To my knowledge no smooth structure has yet been constructed on this automorphism group in general (though it seems to be an interesting question if there is one). For more references let me sneak in my recent paper The Lie group of vertical bisections of a regular Lie groupoid. The references can be found in the introduction together with an explanation of how this is related to the automorphism group.