Learn Hardcore Icelandic: Undirskriftin hans vantar enn á síðunni. - His signature is still missing on the page.

It’s the suffixed definite article. undirskrift = signature; undirskriftin = the signature. Icelandic commonly attaches the definite article as a suffix to nouns.

Should it be undirskriftin or undirskriftina with vantar?

Traditional grammar treats vanta as an impersonal verb that takes the missing thing in the accusative, so: Það vantar undirskriftina or (with fronting) Undirskriftina vantar. In real usage, many speakers also use nominative with a fronted “missing thing”: Undirskriftin vantar (as in your sentence). You’ll hear both; if you want to be conservative, use the accusative.

How does the verb vanta work, exactly?

It’s typically impersonal: there’s no nominative subject.
The person who lacks something is in the accusative: Mig vantar peninga (I need money).
The thing missing is also in the accusative in the neutral pattern: Það vantar undirskriftina (The signature is missing).
Common variants: front the missing thing (Undirskriftina/Undirskriftin vantar).
Forms: infinitive að vanta, present vantar, past vantaði, supine/pp vantað.

Why is hans after the noun (in undirskriftin hans) instead of before it?

With 3rd‑person possessives, Icelandic normally places the possessive after the noun: undirskrift hans / undirskriftin hans = his signature. Preposed hans undirskrift is possible but is typically emphatic or stylistically marked.

Do we need the definite article when there’s a possessor? Is undirskrift hans different from undirskriftin hans?

Both are common:

undirskrift hans (no article) feels a bit more formal/written.
undirskriftin hans (with article) is very common in everyday speech (often called “double definiteness”). Meaning-wise here they both refer to a specific, identifiable signature.

Why not use sinn/sína/sitt instead of hans?

The reflexive sinn/sína/sitt refers back to the grammatical subject of the clause. In Undirskriftin hans vantar…, the verb is impersonal and there is no nominative subject to control a reflexive, so you cannot use sinn. Use hans (his), hennar (her), þeirra (their), etc.

What case is síðunni, and why that ending?

It’s dative singular definite of síða (page): á síðunni = on the page. The preposition á takes:

Accusative for motion/direction: á síðuna (onto the page).
Dative for place/location: á síðunni (on the page). Here we’re talking about a location, so dative is required.

English says “missing from the page.” Why does Icelandic use á síðunni (“on the page”)?

This is a language-specific collocation. With vanta in the sense “be missing,” Icelandic typically indicates the location using á + dative: vantar (eitthvað) á [stað]. Using frá síðunni would suggest movement away/from, which is not how Icelandic frames this idea.

Where does enn go, and can I use ennþá?

enn (still) normally sits after the finite verb or before a prepositional phrase: vantar enn á síðunni.
ennþá is a very common synonym of enn; enn is a bit leaner/formal, ennþá a bit more colloquial. Both fit here: vantar enn / vantar ennþá.

Is the word order here normal? Why does the verb come second?

Yes. Icelandic main clauses are V2: the finite verb comes in the second slot. In Undirskriftin hans vantar enn á síðunni, the first slot is the fronted topic (Undirskriftin hans), and the verb vantar is second.
Other natural variants (all V2):

Það vantar undirskriftina hans enn á síðunni.
Á síðunni vantar enn undirskriftina hans.

How would I turn this into a yes/no question?

Start with the verb (no expletive það):

Statement: Undirskriftin/undirskriftina hans vantar enn á síðunni.
Question: Vantar undirskriftina/undirskriftin hans enn á síðunni?

Are there synonyms for vanta here?

skorta (impersonal similarly): Það skortir undirskriftina (á síðunni).
Periphrastic: Undirskriftin hans er enn ekki á síðunni (His signature is still not on the page).
vanta is by far the most idiomatic in this exact “something is missing” sense.

Is undirskrift the only word for “signature”?

Use undirskrift for the mark/signature itself. undirritun is the act or event of signing (e.g., a treaty signing), not the graphic signature on a page.

Any easy mistakes to avoid in spelling here?

enn (still) vs en (but): double n for “still”.
Dative on page: á síðunni (not á síðuna for this meaning).
Possessive: hans doesn’t inflect; don’t try to agree it with undirskrift.