Can the House Robber problem be solved with a Sliding Window solution?
Yes you can! It's called dynamic programming, though.
Usually, the sliding window technique stores only information local to the window, but to make this work, you should store also the global solution up until the current window position.
What you store in a window (size 2?) is what the optimal solution is if you pick the last house of the window, and what is the optimal solution when you skip the last house of the window.
We call this technique dynamic programming, and not sliding window, although they have much in common.
P.S. this problem is also known as Independent Set in Path Graphs.

I don't know if you call this sliding window, but here is one way of solving it:
def window(data):
    if not data:
        return 0
    pick, skip = data[0], 0
    for window_idx in range(1, len(data)):
        pick, skip = data[window_idx] + skip, max(skip, pick)
    return max(pick, skip)


data = [14, 7, 11, 8, 13, 9, 10, 12]
print(window(data))  # 14 + 11 + 13 + 12 = 50