I have a martial character I want to multi-class into warlock but from what I can see in the book, it looks like they would not receive an arcane focus since they don't get their new class' starting equipment.

How does that work? There's not a price given for the Arcane Focus. Even if there was how would I go about getting one in the case of a warlock? Since the focus would be connected to a being I could not just visit like a Cleric might a temple where would I go to get one if anywhere?

How does a character multiclassing into warlock get a focus? Do multiclassing characters get a free Arcane Focus?

The GM that's running this does not house rule, he's pretty strict about sticking to RAW so please keep answers strictly based on RAW.

You have to buy one, you don't get it for free

Multiclassing does not provide you with the starting equipment for a class:

You don't, however, receive the class's starting equipment

Buying a Focus

But you can buy one! Arcane Foci are available under Adventuring Gear and do have an associated cost:

Item Cost Weight

Crystal 10 gp 1 lb.

Orb 20 gp 3 lb.

Rod 10 gp 2 lb.

Staff 5 gp 4 lb.

Wand 10 gp 1 lb.

Foci and class relationship

This may be have some DM influence, but in general there are no requirements on where you get your Arcane Focus. In fact, the focus is just described as:

An arcane focus is a special 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 the Spellcasting section.

Inherently, the focus just helps you and makes it so you don't require material components (that have no value or aren't consumed.) How you use the focus is up to the character, but it's purpose isn't tied to where you got it - it's tied to how you use it.

No, there is no free equipment when you multiclass unless explicitly stated.

In the PHB it states:

"When you gain a new level in a class, you get its features for that level. You don't, however, receive the class's starting equipment, and a few features have additional rules when you're multiclassing: Channel Divinity, Extra Attack, Unarmored Defense, and Spellcasting."

Think of it this way; if you started as a Warlock then multiclassed into a Paladin, would they suddenly find chain mail under their pillow?

On page 148 of the PHB, under Adventuring gear is a list of items, including Arcane Focus, which run between 5-20 gp. And there is nothing in the description of Warlock that would suggest that the focus needs to be tied to your specific patron.