Answer
$NO_3^-$ has the greater bond lengths.
Work Step by Step
If you draw the lewis structure of $NO_3^-$, you will see that there is 1 NO double bond and 2 NO single bonds. Each oxygen atom on average has 1/3 of a double bond.
If you draw the lewis structure of $NO_2^-$, you will see that there is 1 NO double bond and 1 NO single bond. Each oxygen atom on average has 1/2 of a double bond.
Because double bonds tend to be shorter than single bonds, on average the bond lengths of $NO_3^-$ will be shorter.
