Sharia Watch director Anne Marie Waters suggested "millions" of Britons agree with her view that Islam is evil.
But her candidacy has divided the party and led to reports that 18 out of 20 Ukip MEPs could quit if she wins the leadership.
Former Ukip leader Nigel Farage has also warned that the party will be "finished" if it goes down the road of becoming anti-Islam.
Prompted to address her previous statement that Islam is evil, Ms Waters told BBC Newsnight: "Yes, and I don't see why that's an enormous or outrageous thing to say.
"We ought to be able in this country to say whatever we like about a religion and the problem we have got is that we pussyfoot around, we spend so much time agonising over not saying the wrong thing, and this is what's putting the public off.
"But this is how millions of people in this country feel and they are waiting for someone to articulate it for them."
Ms Waters, who has co-founded a UK wing of German anti-Islam movement Pegida with Mr Robinson, said she would leave it up to party members to decide if the former EDL leader could join Ukip if he expressed that desire.
"There is a leader's discretion, I think I would leave it up to party members," she said.
"I wouldn't lift the ban on groups such as the BNP."
But one of her main rivals for the leadership, deputy leader Peter Whittle, warned: "It's not up to whether the members want him or not, it's in the constitution, it's as simple as that, it is not going to do any good for this party if those sort of people start to join this party."
Ms Waters hit back, saying: "A lot of people around the country support those sort of people."
Ukip will declare the winner of its leadership election at the party's conference in Torquay on September 29.
The party endured a disastrous general election which left it with no MPs and only 1.8% of the UK vote.