It has installed 60 fan heaters to keep employees warm, and said the Health and Safety Executive was satisfied with what it has done.
But Ms Bee, the industrial officer for transport at PCS, said the fan heaters were only 30cm (12in) high and not good enough for the 15-floor office.
She said they are "the type of fan heater that you would expect to see in a home heating one room".
"The boiler went down last Thursday evening, and there was no communication to staff in advance of them arriving on site on Friday morning."
She added that a lack of hot running water in the toilets meant staff were having to go to the kitchen to wash their hands.
"It's not lost on us that the DVLA saw the biggest Covid outbreak in the government estate, and we now see reckless actions to keep the site open, despite the fact that staff aren't able to wash their hands after using the toilet facilities, in hot water."
Workers at the DVLA site went on strike in 2021 after hundreds of Covid cases at the office in the Morriston area of Swansea.
A DVLA spokesperson said: "The health, safety and welfare of our staff is a priority and we have consulted the Health and Safety Executive who are satisfied with the measures we have taken.
"Temporary heating systems have been installed to ensure staff who are still working onsite are kept warm. Hot water and hot food is also available."
What is the law on office temperature?