Ich prüfe Bestellnummer und Zustellnachweis.

Prüfe is the 1st person singular present-tense form of the verb prüfen (to check, to examine, to verify).

The verb is conjugated like this in the present tense:

• ich prüfe = I check
• du prüfst = you check
• er/sie/es prüft = he/she/it checks
• wir prüfen = we check
• ihr prüft = you all check
• sie/Sie prüfen = they check / you (formal) check

So Ich prüfe simply means I check or I am checking.

A learner will often expect something like:

• Ich prüfe die Bestellnummer und den Zustellnachweis.

That version is very natural in standard German.

The article-less version:

• Ich prüfe Bestellnummer und Zustellnachweis.

sounds more like compressed, business, checklist, interface, or note style. German often drops articles in headings, forms, bullet points, system messages, and procedural language.

So:

• with articles = more complete, normal full-sentence German
• without articles = more concise, label-like, procedural, or bureaucratic style

They would be:

• die Bestellnummer
• den Zustellnachweis

So the full sentence would be:

• Ich prüfe die Bestellnummer und den Zustellnachweis.

Why those forms?

• Bestellnummer is feminine: die Bestellnummer
• Zustellnachweis is masculine: der Zustellnachweis

Because these nouns are the direct objects of prüfen, they are in the accusative case:

• feminine accusative: die
• masculine accusative: den

They are in the accusative case, because they are the things being checked.

In the sentence:

• Ich = subject
• prüfe = verb
• Bestellnummer und Zustellnachweis = direct objects

If the articles were shown, you could see the accusative more clearly:

• die Bestellnummer (feminine accusative)
• den Zustellnachweis (masculine accusative)

Without articles, the noun forms themselves do not change here.

Because German very commonly forms compound nouns by joining words together.

Here is how they break down:

• Bestellnummer = Bestellung (order) + Nummer (number)
  → order number
• Zustellnachweis = Zustellung (delivery) + Nachweis (proof/evidence/confirmation)
  → proof of delivery

This is one of the most important patterns in German vocabulary. English often uses several separate words; German often combines them into one noun.

In German, all nouns are capitalized, not just proper names.

So in this sentence:

• Ich is capitalized because it is the first word of the sentence
• Bestellnummer is capitalized because it is a noun
• Zustellnachweis is capitalized because it is a noun

This is standard German spelling.

It can mean either one, depending on context.

German does not usually use a special progressive form like English am checking. The simple present tense often covers both meanings:

• Ich prüfe ... = I check ... / I am checking ...

So the exact English translation depends on the situation.

This follows the normal basic German main-clause pattern:

• subject + verb + rest

So:

• Ich = subject
• prüfe = verb
• Bestellnummer und Zustellnachweis = object(s)

German main clauses usually put the finite verb in second position. Here, the subject comes first, so the verb comes right after it.

Because both nouns are being checked by the same verb prüfe.

So German, like English, can use one verb with two coordinated objects:

• Ich prüfe Bestellnummer und Zustellnachweis.

That means:

• I check the order number
• and I check the proof of delivery

There is no need to repeat the verb unless you want extra emphasis:

• Ich prüfe die Bestellnummer und prüfe auch den Zustellnachweis.

That version is grammatical, but usually less natural unless you want to stress each action separately.

It is understandable, but in ordinary full-sentence German, many speakers would prefer:

• Ich prüfe die Bestellnummer und den Zustellnachweis.

The version without articles feels more natural in contexts like:

• a workflow description
• software text
• a checklist
• office or logistics language
• short notes

So the original sentence is not wrong, but it has a concise procedural style rather than a fully neutral conversational style.

Yes, if you were talking about more than one order number or more than one proof of delivery.

For example:

• Ich prüfe die Bestellnummern und die Zustellnachweise.

That means I check the order numbers and the proofs of delivery.

Singular:

• die Bestellnummer
• der Zustellnachweis

Plural:

• die Bestellnummern
• die Zustellnachweise

So in your original sentence, both nouns are most naturally understood as singular.