House Mormont is the closest Westerosi house that allows women a significant role as war leaders and rulers. Although I am violating the twenty-year rule here, both Dacey and Maege Mormont played a crucial role in Stark's abortive attempt at independence. It's not though that Bear Island is a matriarchal paradise, it's more reflective that particular circumstances, the frequent Ironborn raids in particular, which have meant women play a more equal role than on the continent

House Martell might also be considered similar, but I would argue they are not a true Westerosi house despite their sigils. Martell political culture is a hybrid of native Dornish precepts and the Rhoynar of Essos, with a heavy emphasis upon the latter.

You mean Houses where the lineage always passes to a daughter before a son? I do not believe so, the Rhoynish Dornish Houses are probably the closest you're going to get.

Speaking to your follow up question of informal power, no better example than House Mormont in the North exists. Both girls and boys are trained from childhood in battle so as to protect themselves from the not infrequent Wildling raids. The seat of House Mormont is decorated with a woman nursing a child with one hand and holding a battleaxe in the other. This plainly illustrates the importance women play in Bear Island's function.

Interestingly, the current head of House Mormont is Maege Mormont with the current heir one of Maege's daughters. There are recent examples of the Mormon women fighting alongside men in the Northern Army.

In a great anecdote, a king recently demanded fealty from Bear Island and instead of receiving a response back from a maester or other similar castellon, was greeted instead with a denial from 10 year old Lyanna Mormont insisting that the only king for Bear Island was a Stark.