How can I play a bot with expansion cards in Dominion Online?
You have to purchase a subscription to the expansion sets. While the base-game cards are free, the expansion cards are not. After you log in, go to the Store tab and you shall find two different types of subscriptions available:


Silver, which includes the following sets: Intrigue, Seaside, Prosperity, Cornucopia, Hinterlands, Guilds
Gold, which includes all sets from the Silver subscription, plus Alchemy, Dark Ages, Adventures, Empires


These are per-day subscriptions, not permanent purchases, but you can define the expiry day of your subscription; e.g., if you would like to test the expansions, you can even subscribe for a day or week. Note that there is a transaction fee to be paid as well (0.35 EUR, as of 2017-04-08).

Note that you need to purchase the expansions to open a new table on your own. This, naturally, includes playing alone against a bot. Nevertheless, you can always join for free any table, even with expansion cards, as long as the table administrator has paid for them.