Can I load a hand crossbow if I am using a shield and the crossbow? Would I be able to get multiple attacks with Crossbow Expert if I am using a shield and a hand crossbow?
The errata added “(you need a free hand to load a one-handed weapon)” to the Ammunition weapon property (PHB p. 146). Thus, you wont be able to carry a shield and load your weapon.
As a DM myself, I would allow you to load your weapon if you posses the War Caster feat (PHB p. 170):
You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.
PHB p. 203, SOMATIC (S) says
Spellcasting gestures might include a forceful gesticulation or an intricate set of gestures. If a spell requires a somalic component, the caster must have free use of at least one hand to perform these gestures.
As I said above, I would houserule that the War Caster feat enables you to also load ammunition to a one-handed weapon while wielding a shield. Doing wild gestures with weapons in the hand to manipulate magic should have the same prerequisite as loading ammunition to one-handed range weapon (the hand-crossbow, sling and blowgun are the only one handed range weapons).
Drawing or stowing away your shield is an action, as listed in the "getting into and out of armor" (PHB p. 146).
No, you can't load a hand crossbow while holding a shield. The errata added a new condition to the Ammunition property:
Loading a one handed weapon requires a free hand.
A hand holding a shield is not a free hand, so you can't load your hand crossbow while using a shield. And since you can't load your hand crossbow, you can't make multiple attacks.
As far as D&D 5E is concerned
The errata'd rules prohibit it. You need a free hand to load a crossbow.
In real life (yes, I know :-)
It is possible to reload a hand crossbow with a shield. I've done it. It was in response to exactly this question when playing 1st edition AD&D - yes, this question is that old.
The type of shield we had one with a sleeve and a handle. You slipped your forearm into the sleeve and held on to the handle (i.e. it was not a centre-boss shield). This left the fingers and thumb able to wiggle a little bit - enough to be able to grab bolts from a bandolier and load them into a handbow. Cocking the handbow was done by hooking it into a hook sewn into clothing.
As an aside, we also tried it with small weapons in the second hand, like a dagger. Like the shield, these left the fingers and thumb able to wiggle enough to do simple tasks.
The bow was pretty low-power, but could punch bolts into a pumpkin at around 8m, so I think it was close to what D&D calls a "hand crossbow".
However, "possible" is not the same as "easy". Some of my D&D group were actual weapon users (some fencing, one sword-and-shield) - they were all dubious whether it was possible under combat conditions.
I think a reasonable house rule is: If you take Crossbow Expert feat then you can reload a hand crossbow while using a shield, as long as you've paid a bit of gold for a chest-mounted bandolier.
If you take the Crossbow Expert feat to ignore the loading quality then yes, you can.
You ignore the loading quality of crossbows with which you are proficient.