It is the latest show of unrest at the human rights organisation over a cost-cutting programme that the union says puts up to 40 jobs at risk.
Unite said its members at AIUK, which total about 145 out of 200 employees, want Allen to take responsibility for what it called a "catalogue of poor strategic and financial decisions", which led to notices of possible redundancy being issued to 70 members of staff on Wednesday.
The union said her leadership "lacks transparency, accountability and long-term vision".
Staff at AIUK, based in Shoreditch, east London, have already taken two days of strike action in protest over a lack of consultation with the union, Unite said.
Unite members believe the organisation’s work on human rights will be damaged by cuts totalling £2.5m, which are being made so that AIUK can contribute more money to its international headquarters over the next 10 years with a view to expanding the organisation globally.
AIUK said in a statement that Allen had the continued confidence of its board.
Sally Kosky, Unite’s national officer for the not-for-profit sector, said: "Staff are concerned that Kate Allen’s proposed cuts will seriously damage the UK section’s ability to operate effectively and to provide strong campaigning and fundraising support for the global movement in the longer term.
"Our members have lost confidence in Kate Allen’s ability to guarantee a viable future for AIUK and its human rights work, and are calling on her to resign immediately."
AIUK said in a statement: "This vote demonstrates a strength of feeling that is completely understandable.
"Kate, the senior management team and the board all understand the impact of the cost-saving programme on staff. They are determined to make the consultation over restructuring and redundancies genuine and will ensure that the process is fair and transparent.
"Unite says the cuts are due to the speed at which senior management intends to increase contributions to Amnesty’s international headquarters, yet it was the highest decision-making body of the Amnesty movement, the International Council Meeting, that made decisions on how much national sections like Amnesty International UK should contribute.
Staff at Amnesty International’s headquarters, based in Islington, are involved in a separate dispute with their management and took part in a strike over changes to its redundancy policy.