n=9;

γsol = (((E^(c[1])) ((-1+(E^(c[2]))) c[1]-c[2])+c[2])/((-1+(E^(c[1]))) c[2] (c[1]+c[2])));

Series[γsol /. c[i_] :> c[i]h, {h,0,n}]

eqns1 = {(-(E^(c[1])) (c[1]+c[2]) (1+α c[1]+γ c[2]) (1+β c[1]+δ c[2])+c[2] (1+δ c[2]+α c[1] (1+δ c[2])+γ (c[1]+β ((c[1])^2)+c[2]+δ c[1] c[2]+δ ((c[2])^2)))+(E^(c[1]+c[2])) c[1] (1+δ c[2]+β (c[1]+c[2]+γ ((c[2])^2))+α c[1] (1+δ c[2]+β (c[1]+c[2]))))/(c[1] (c[1]+c[2]) (1+α c[1]+γ c[2]) (1+β c[1]+δ c[2]))==0,((E^(c[1])) (c[1]+c[2]) (1+α c[1]+γ c[2]) (1+β c[1]+δ c[2])-c[2] (1+δ c[2]+α c[1] (1+δ c[2])+γ (c[1]+β ((c[1])^2)+c[2]+δ c[1] c[2]+δ ((c[2])^2)))-(E^(c[1]+c[2])) c[1] (1+δ c[2]+β (c[1]+c[2]+γ ((c[2])^2))+α c[1] (1+δ c[2]+β (c[1]+c[2]))))/(c[2] (c[1]+c[2]) (1+α c[1]+γ c[2]) (1+β c[1]+δ c[2]))==0} /. {α|β|δ -> 0}

Reduce[eqns1, γ]

γs = Sum[(Subscript[γ, 10i+j])c[1]^i c[2]^j h^(i+j), {i, 0, n}, {j, 0,n}]+O[h]^(n+1)

eqns2=(eqns1 /. {γ -> γs, c[i_] :> h c[i]} // Simplify // Normal // Collect[#, h]&);
vars=Union[Cases[eqns2, (Subscript[γ, _]),Infinity]];
psol=SolveAlways[eqns2, {c[1], c[2], h}]

γs /. psol // Simplify

Series[γsol /. c[i_] :> h c[i], {h,0,6}]

vars

(1/6) c[2] (2 c[1]+c[2]-3 (c[1]+c[2]) (Subscript[γ, 0])+6 c[2] (Subscript[γ, 0]^2)-6 c[2] (Subscript[γ, 1])-6 c[1] (Subscript[γ, 10])) /. {(Subscript[γ, 0])->(1/2),(Subscript[γ, 10])->((c[1]+2 c[2]-12 c[2] (Subscript[γ, 1]))/(12 c[1]))} // Simplify

Simplify[eqns2 /. psol]