In building a Cleric/Wizard, is there a mechanism by which I could buy or craft a shield that is an arcane focus with a holy symbol emblazoned on it?
It's up to your DM, but there's not a rule forbidding it.
Per Arcane Focus (page 151 5e PHB) emphasis mine,
Arcane Focus. An arcane focus is a special item — an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item— designed to channel the power of arcane spells. A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in chapter 10.
Per Spellcasting (for multiclassing) (page 164 5e PHB),
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
There is nothing in the rules preventing you from combining the focuses from various classes, such as setting an arcane-focus orb into the shield that also bears your holy symbol.
Let's look at the play mechanics of combining an arcane focus and a shield with a holy emblem on it. All quotes from 5e PHB.
Per Shield (page 144),
A shield is made from wood or metal and is carried in one hand. Wielding a shield increases your Armor Class by 2.
Per Material (M) (page 203),
Casting some spells requires particular objects, specified in parentheses in the component entry. A character can use a component pouch or a spellcasting focus (found in chapter 5) in place of the components specified for a spell. ... A spellcaster must have a hand free to access these components, but it can be the same hand that he or she uses to perform somatic components.
Per War Caster feat (page 170),
You can perform the somatic components of spells even when you have weapons or a shield in one or both hands.
Therefore you must have one free hand to access the arcane focus. As a cleric, that hand can also be the hand that hold the shield. The War Caster feat makes it seem like using a hand that is holding a shield to also perform the somatic components of a spell is not an inherent ability for non-holy-symbol-focus-based classes. So, holding a shield in one hand and removing the orb from the shield with the other is a completely legal move.
Even when using weapons, you are mostly fine. Per Interacting with Objects Around You (page 196),
Here are a few examples of the sorts of thing you can do in tandem with your movement and action: draw or sheathe a sword ... pick up a dropped axe ...
So, in your turn, you could draw your weapon, and either attack or cast a Cleric spell. Then the next turn you could drop your weapon for free and draw your arcane focus and cast a spell from either class. The following turn you could drop your arcane focus and pick up your dropped weapon, then attack. You could essentially drop-juggle your arcane focus and weapon, which looks incredibly silly but is completely legal as far as the rules go, as long as your movement always takes you back to the same place that you are dropping your items. That last bit is going to be picked up by your DM, and you may find that another creature snags your dropped loot, which is the natural counter to that tactic.
Still, you would have the same effect with a component pouch and a shield. The only difference is that you wear your component pouch, so you don't have to drop anything, since you could either draw or sheathe your weapon as your free object interaction every turn.
One last thing, the arcane focus is "some similar item designed to channel the power of arcane spells". There's nothing preventing you from trying to get a shield designed to channel the arcane. At the end of the day, it's just up to your DM, and is mostly flavoring.
Here's some other relevant links:
\$\begingroup\$ Sure, which is why "Ask your DM" is usually the answer to the question "Can I combine the spellcasting focuses from class X and class Y?" \$\endgroup\$ Jul 10, 2017 at 20:34
Shield is not one of the available options for an arcane focus. You're 100% in house-rule territory.
One of the reasons for allowing Clerics (and Paladins) the use of a holy symbol on a shield is because it's not fundamentally different from any other use of a holy symbol - wearing a holy symbol is sufficient (Holy Symbol, PHB p151).
Other focuses must actually be held in hand (Material (M), PHB p203).
\$\begingroup\$ Druids can use a totem, which I have interpreted as a type of amulet made out of fur and bones. It says nowhere that I can see that you have to hold your focus in your hand. Theoretically the crystal and orb arcane focuses can also be worn around the neck. \$\endgroup\$ Jul 10, 2017 at 23:09
1\$\begingroup\$ @emerald serpent His quote and link is from D&D Beyond, an official WotC online resource that has been updated more recently than the PHB. \$\endgroup\$– T.J.L.Jul 22, 2017 at 8:53
3\$\begingroup\$ Yep. The PHB has also been updated to reflect the same change, as can be seen from the PHB errata regarding material components: media.wizards.com/2017/dnd/downloads/PH-Errata.pdf \$\endgroup\$– V2Blast ♦Jan 31, 2018 at 5:12
RAW rule: There is a specific rule on Holy symbols that allows it to be used as spell focus while it is engraved on a Shield. Specific beats general, so that use of a Holy Symbol as a spell focus is permitted but NOT an arcane focus. PHB p151 Holy symbol description.