Does Smite work in beast form? I have a plan for a half orc Paladin/Druid build, but I want to make sure this aspect of the build works before I play the character.
Yes, Divine Smite will work while Wild Shaped. Wild Shape allows you to use your class features:
You retain the benefit of any features from your class, race, or other source and can use them if your new form is physically capable of doing so.
And Divine Smite works like this:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target, in addition to the weapon's damage.
Divine Smite requires a melee weapon attack. Every beast type creature that I can think of has at least one attack that is a melee weapon attack. For example, the Wolf:
Bite. Melee Weapon Attack: +4 to hit, reach 5 ft., one target. Hit: 7 (2d4 + 2) piercing damage. If the target is a creature, it must succeed on a DC 11 Strength saving throw or be knocked prone.
So you can use your Divine Smite with the melee weapon attacks provided by your Wild Shape form.
The rule says it applies when you make a "melee weapon attack," so unless your beast form is using a melee weapon, I would say "no."