In Arianne’s TWOW chapters she says that “if (Daenerys) hopes to sit the Iron Throne, she must have Sunspear”. Is this true? None of the claimants in the War of Five Kings seemed particularly interested in Dornish support and we know they’re militarily under-powered from Doran.

I think it’s true in the following sense: 

• The Westerlands, North, Iron Islands, Riverlands, Vale, Stormlands have strong anti-Targaryen leanings coming out of Robert’s Rebellion. They’re unlikely to lend any assistamce to Dany, especially straight off the boat.
• The Reach is committed to Tommen, whether through Margaery, land distributions after Blackwater, coronation oaths, etc. There are probably some Targaryen loyalists left among the lords of the Reach, but they’re isolated and unlikely to be free to act. 
• Dorne is the only intact Targaryen loyalist kingdom in Westeros. Not only can they give Dany 25,000 spears, but they can give her the legitimacy of a kingdom’s worth of (princes and) lords bending the knee and acclaiming Dany the rightful Queen of Westeros.