The general algebraic question is multifaceted and can be quite complicated. Some depend on vector space, some on extension field properties.
As mentioned in the comments checking the property can be simpler.
I answered a related question Examples of multi output bit balanced Boolean functions
Nyberg’s articles mentioned there are
K. Nyberg, Differentially uniform mappings for cryptography, 1993 and
K Nyberg, Perfect nonlinear S-boxes, 1992
both easily locatable on google scholar.
Edit: The keccak $\chi$ maps $\{0,1\}^5$ to itself.
I will use $a_i$ as input and $A_i$ as output variables as in the edited question.
Counting indices modulo 5, if there is no $i$ such that $(a_i,a_{i+2})=(0,1)$ then $\chi$ has a fixed point for that input. Let $W=\{i: (a_i,a_{i+2})=(0,1)\},$ then the general mapping just inverts the bits belonging to $i.$
Note that the sets $J_i,J_j$ where $J_i=\{i,i+2\}$ are disjoint except when $j=i+2$ or $i=j+2.$ So there is no ambiguity for determining the inverse unless we are in this special case, thus the inverse exists except in this special case. But even in this case the patterns
$(a_i,a_{i+2},a_{i+4})$ which result in bitflips are unambiguous.
If $(a_i,a_{i+2},a_{i+4})=(1,0,0)$ then $a_{i+1}$ will be flipped but not $a_{i+3}$. So $A_{i+1}=1\oplus a_{i+1},$ and $A_{i+3}=a_{i+3}.$
If $(a_i,a_{i+2},a_{i+4})=(1,0,1)$ then $a_{i+1}$ will be flipped but not necessarily $a_{i+3}$, that will depend on the value of $a_{i+6}=a_{i+1}$. But that bit is not impacted by the previous argument since $J_i$ and $J_j$ are disjoint if $i=j+1\pmod 2.$
So a unique inverse mapping exists.
Remark: In general going between "basis independent" extension field formulations of permutations vs "basis dependent" bit vector permutations is hardly straightforward. I don't see an immediate basis independent extension field formulation for this permutation, and as pointed out in the comments to the question such formulations obtained (say) by Lagrange interpolation, can be quite complicated and high degree.