Web interface for GAP (or other computer algebra system dealing with finite groups)?

You can get at GAP through http://sagemath.org, which has a perfectly fine web notebook interface.

There is the Magma calculator which can be used to do calculations in finite groups.

One problem is that you have to type in all of your input before executing it, but with practice you can do quite complicated calculations.

For example, you can carry out the calculation from my answer to this question (which was actually about infinite groups). Typing in the following code

G<x,y,z>:=Group<x,y,z|x*y^-1*x^-1=z^2*y, x*z^-2=z^2*x, x*y*x^-1*y=z^2,
  y^2*x*z=z*x >;
K<a,b,c,d> := sub<G | x^2, z^2, x*z*y^-1, y^2>;
Index(G,K);
Rewrite(G,~K);
K;
Transversal(G,K);
PK, phi := ElementaryAbelianQuotient(K,2);
Order(PK);
K2 := Kernel(phi);
Index(K,K2);
T2 := Transversal(K,K2);
exists{k : k in T2 | (x*k)^2 in K2 };
exists{k : k in T2 | (y*k)^2 in K2 };
exists{k : k in T2 | (z*k)^2 in K2 };

results in the output:

4
Finitely presented group K on 4 generators
Index in group G is 4 = 2^2
Generators as words in group G
    a = x^2
    b = z^2
    c = x * z * y^-1
    d = y^2
Relations
    (c^-1, a) = Id(K)
    (a^-1, b) = Id(K)
    (a^-1, d^-1) = Id(K)
    (d^-1, b^-1) = Id(K)
    (b, c) = Id(K)
    d * c * b^-1 * d^-1 * c^-1 * b^-1 = Id(K)
    b^-1 * a * c^-1 * a^-1 * b * c = Id(K)
{@ Id(G), x, y, z @}
Mapping from: GrpFP: G to {@ Id(G), x, y, z @}
16
16
false
false
false

There exists a Jupyter kernel for GAP, see https://github.com/gap-packages/jupyter-kernel-gap

A simple way to get this actually running is through SageMath: if you have a recent beta(!) version of SageMath installed (or wait until 8.0 gets released), you can run sage -i gap_jupyter to install that kernel. At that point, you start Jupyter with sage -n jupyter and then create a New GAP notebook using the Jupyter menu.