Learn French: Faute d'enclume, le forgeron a dû abandonner son projet. - For lack of an anvil, the blacksmith had to abandon his project.

After the expression faute de (for lack of), we usually drop indefinite and partitive articles (un, une, du, de la, des). We just use de directly followed by the noun. Because enclume starts with a vowel, de contracts to d'.

How does faute de compare to par manque de? Could I use that here?

Yes, par manque d'enclume would mean exactly the same thing. Faute de is slightly more formal and elegant, but both are excellent ways to express that the absence of something prevented an action.

The lesson mentions faute de can be followed by an infinitive. How would that look in this sentence?

To use an infinitive here, you could say Faute d'avoir une enclume... (For lack of having an anvil...). Notice that when you add the verb avoir, the article (une) comes back because the noun is now the direct object of avoir, rather than directly attached to faute de.

Why is the verb a dû in the passé composé instead of the imparfait (devait)?