Does the problem persist with this modification
eqns = {(1-r^2) r D[c[z,r],z]==1/10(r D[c[z,r],r,r]+D[c[z,r],r]+D[c[z,r],z,z])-1/100 c[z,r],c[0,r]==0,
(D[c[z, r], r] /. r -> 0) == 0, c[z, 1] == 1, (D[c[z, r], z] /. z -> 1) == 0};
u = NDSolveValue[eqns, c, {z, 0, 1}, {r, 0, 1}]
No. That doesn't appear to work either. And neither does this
eqns = {(1-r^2) r D[c[z,r],z]==1/10(r D[c[z,r],r,r]+D[c[z,r],r]+D[c[z,r],z,z])-1/100 c[z,r], c[0,r]==0,
Derivative[0, 1][c][z, 0] == 0, c[z, 1] == 1, Derivative[1, 0][c][1, r] == 0};
u = NDSolveValue[eqns, c, {z, 0, 1}, {r, 0, 1}]
So I've exhausted my tips. Sorry. Hopefully someone better will be able to see what you need to do.