Why is dinitrogen tetroxide a planar molecule?
I think that one would expect that a stable form of the molecule might be planar and have D2h symmetry.  

Some support for the claim: 


Early crystal structures suggested that the ONO bonds on either side
of the N-N bond are easily modeled by low level DFT
(doi.org/10.1039/C19660000082).
More recently, simulations of the trans and cis N2O4 structure show
planarity when optimized at B3LYP/ aug-cc-pVTZ (aka, pretty good not
amazing) See here. 


The simulations also suggest that the barrier to isomerization is low, meaning that while the stable structures are planar, there are low barrier (low temperature) out of plane rotations possible.

Summary:


Just because something has sigma bonds doesn't imply that it will immediately be non-planar 
yes the molecule can form a conformer