In the card crafting UI, is it possible to search for cards that I do not own
No, the tools are too limited.
You have only 5 filter options available:
- the mana gem cost: 0, 1, 2, 3, 4, 5, 6 or 7+
- the search box (finds all cards with the word, so "murloc" finds Hungry Crab)
- the set (more on this later)
- Show Only Golden Cards
- Include Soulbound Cards (shows more cards)
If you want to find cards you can craft which you are not yet able to play, you need to do the following:
- Set filters to: all mana gems, empty search box, Expert Set, do not Show Only Golden Cards, do not Include Soulbound Cards.
- examine each card. If you have 0, the card will appear blue. If you have 1, there will not be a x2 tag. There should be a x2 tag on all the non-golden cards unless (a) the card is legendary (you only get one) or (b) you have additional gold versions of the card bringing you to 2 total.
- Change filter to Promo Set
- repeat step 2
There is no need to check the Basic Set, Reward Set or the Naxxramas Set as all non-golden cards in these sets are Soulbound (cannot be crafted). Do check the Include Soulbound Cards filter if you are examining these sets.
With the new patch 4.2.0 (released in middle March 2016) many new filtering options have been added. If you click on the crafting button and type the following in the search box:
owned:0-1
then you'll only see all craftable cards for which you only own 0 or 1 cards. If you also want to include uncraftable/adventure cards select the "Include uncraftable cards" checkbox.
More details about all the new filtering options are detailed in this Hearthstone blog article.