Mam siedem awiz.

Because Polish often drops subject pronouns when they are obvious from the verb ending.

Mam already means I have, so ja is not necessary.

• Mam siedem awiz. = neutral, normal
• Ja mam siedem awiz. = more emphatic, like I have seven notices

Mam is the 1st person singular present tense of mieć = to have.

So:

• mam = I have
• masz = you have
• ma = he/she/it has

In this sentence, mam tells you both the person and the number: it is I, singular.

The basic dictionary form is awizo.

After numbers 5 and above, Polish usually puts the counted noun in the genitive plural.
The genitive plural of awizo is awiz.

So:

• jedno awizo = one notice
• dwa awiza = two notices
• siedem awiz = seven notices

That is why the sentence uses awiz.

Awiz is genitive plural.

This is one of the standard Polish number patterns:

• after 1: singular
• after 2, 3, 4: usually nominative/accusative plural
• after 5+: genitive plural

So in Mam siedem awiz, the noun after siedem must be awiz.

As a whole, siedem awiz functions as the direct object of mam, so it is an accusative object in the sentence.

But inside that phrase, the noun itself appears in the genitive plural because of the numeral siedem.

This is a very common Polish pattern and can feel unusual to English speakers, because English does not change the noun after numbers in this way.

No. Polish numbers behave in groups.

A useful basic pattern is:

• 1
  • singular
    
    • jedno awizo
• 2, 3, 4
  • plural
    
    • dwa awiza
    • trzy awiza
    • cztery awiza
• 5 and higher
  • genitive plural
    
    • pięć awiz
    • sześć awiz
    • siedem awiz

So siedem belongs to the 5+ group.

The singular form is awizo.

Useful forms here are:

• singular: awizo
• plural used after 2, 3, 4: awiza
• genitive plural used after 5+: awiz

Examples:

• Mam jedno awizo.
• Mam dwa awiza.
• Mam siedem awiz.

Yes. Awizo is neuter.

You can see that from forms like:

• jedno awizo
  The numeral jedno is neuter, so it agrees with a neuter noun.

This matters especially when you use one or adjectives with the noun.

Because Polish does not have articles like English a/an and the.

So Mam siedem awiz does not need any extra word before the noun.
Whether English translates it as I have seven notices, I have seven delivery notices, or in some context I have the seven notices depends on the situation, not on an article in Polish.

Mam siedem awiz is the most neutral word order.

Polish word order is more flexible than English, so other orders are possible, but they usually change the emphasis:

• Mam siedem awiz. = neutral
• Siedem awiz mam. = emphasis on seven notices
• Ja mam siedem awiz. = emphasis on I

So the original sentence is the most natural basic version.