So you don’t have a Tarot deck. That’s okay: House of Cards works just fine without one, though you have to do a slight amount of preparatory work. The two places it impacts the game is during character generation and for resolving tasks.
Character generation requires you to determine which Archetype you’ve been invested by, but that doesn’t mean you have to draw a card. As always, just choosing one of the Major Arcana from a list (omitting the World) is an option, and there are plenty of pictures of many different decks online for you to peruse in picking your correspondences. To get a random draw, use an online virtual die roller to randomly pick a number from 1-21 (the number of playable Archetypes) and scan down the list that many places.
By contrast, you’ll have to deal with task resolution much more often than character creation, but the solution is pretty easy: instead of having the Minor Arcana on hand, substitute a deck of playing cards with two extra jokers. Label the jokers as the Pages for each suit, and you’re good to go.