They did; the Palace was hit by bombs on several occasions, most notably on September 13th 1940. The King and Queen were present and not taking shelter at the time; the Queen wrote to her mother-in-law about the experience:

"It all happened so quickly that we had only time to look foolishly at each other when the scream hurtled past us and exploded with a tremendous crash in the quadrangle," she wrote.

While her "knees trembled a little bit", she was "so pleased with the behaviour of our servants", some of whom were injured as one bomb crashed through a glass roof and another pulverised the palace chapel.

It's unlikely the Palace was specifically targeted, though. The vast majority of bombing during the war was highly inaccurate, and German attacks were primarily economic in nature, aimed at port facilities in London. If there was intent to hurt morale by bombing the Palace the effect was quite the reverse. Despite the popular images of cheering crowds greeting Royal visits, "Keep Calm and Carry On", "Britain Can Take It" and such ("The Myth of the Blitz" as per Angus Calder's book title), there wasn't unified stoicism. There was disquiet as the lower class tightly packed housing around the docks was disproportionately affected by raids and East End residents felt 'the toffs' weren't being hit; as Harold Nicolson wrote in his diary: "Everybody is worried about the feeling in the East End, where there is much bitterness. It is said that even the King and Queen were booed the other day when they visited the destroyed areas (...) if only the Germans had had the sense not to bomb west of London Bridge there might have been a revolution in this country. As it is, they have smashed about Bond Street and Park Lane and readjusted the balance." The death of the King may have had an impact on morale (though may also have strengthened resolve or desire for retribution) but the propaganda opportunity of his heroic survival was obvious, journalists were rapidly invited to come and inspect the bomb damage and as the Queen famously said, "I'm glad. It makes me feel I can look the East End in the face."