The Ikko-Ikki monks were always a thorn in people's side, but they were never a contender for control of Japan. They were a regional religious sect that appealed to the lower classes, a powerful one at that, but they lacked the universal appeal needed for a political movement to grow and replace the feudal mechanical structure of power. It's a lot harder to take control of a region from outside the normal channels. While Daimyo could defeat their rivals and assimilate their lands and forces, the Ikko-Ikki would have had to convert people to their way of thinking, which is far more difficult and realistically impossible without a complete breakdown in people's trust of existing power brokers.

Turnbull, The Samurai

I would say no.

Ikko Ikki was not a single united faction. Any local uprising of peasants and/or ji-samurai that had Ikko Buddhist ties is called Ikko Ikki.

The Ikko Ikki controlled Kaga, Echizen, parts of Ise and fringes of Mikawa, but they were not a unified force (take a look at the map and that much should be obvious), and there were not inconsiderable infighting.

Due to their nature, there's also no way samurai elites would bow to them, so the Ikko would need to exert direct control over the area and not just negotiate surrender and switching sides as joudai. And the kuge in Kyoto would never lend them any sort of legitimacy.

Though frankly after Anegawa, the only serious contender for control of Japan was Oda and allies, not counting what happened later to lead up to Sekigahara.

I'd also like to take the opportunity to add another very related question that I've always been meaning to ask but thought it was too small for its own thread.

In A.L. Sadler's history book Shogun, there is a chapter called Ieyasu Quells the Monto Sect in which Tokugawa Ieyasu helps out Nobunaga eradicate the rebellious groups. However I have not found any other mention of the Monto Sect in any other material. I was curious as to what denomations over history had led religious Buddhists to such a belligerent path but my research yielded nothing. Was there ever such a Monto Sect? What groups did Sadler refer to?