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As a non-native English speaker I struggle with Greek prefixes. Am I allowed to use just normal English numbers in place of them? Is it natural? Or do I have to learn how those Greek prefixes work at the end of the day?

Mathematicians do use this form for bigger numbers. The Wikipedia article Heptadecagon currently contains the phrase "a regular 51-gon, 85-gon or 255-gon and any regular n-gon with 2h times as many sides. "

And in that context, you may find mathematicians using the form for smaller numbers: in an article about polygons of different sizes, I would not be surprised to meet "5-gon" or "8-gon".

But outside mathematics, I've never heard anybody say "eight-gon" or any of the others.

Short answer: no.

Longer answer: you will probably be understood, but people will think it's strange. Almost all words in English have roots in other languages. For these words, you have correctly identified Greek as the original language. Curiously, you have taken exception with the prefixes, but not the suffix: -gon, which is also Greek.

As a more interesting case, triangle uses the -angle suffix, which comes from Latin.
_{( tri- is both a Greek and Latin prefix.)}

Older Germanic languages had very few nouns and adjectives to describe geometric shapes. English had native words for a circle, trendel and heing, an arch boga, cf. Ger. Bogen, and words for three- and four-cornered, _ þriscíte_ and, variously spelled, feoðerscéte , feðerscíte , fiðerscýte , fyðerscýte.

Square, from Old French esquire, first entered English as a carpenter’s square mid 13th c., a square shape/area in the late 14th c., and finally as a rectangle with equal sides in the 1550s. This would suggest that a carpenter couldn’t use a square to draw the eponymous shape for well over a century.

Triangle was borrowed directly from French in the late 13th c.

The most common names for polygons with 5–8 sides entered English from Greek via Latin in the 1560s. This tells you a great deal about the systematic study of geometry in England as pentagonon and octagonon lost their endings to become what we call them today. You’re basically stuck with them. Such bizarre grecogermanic hybrids like *five-gon do not exist.

Other Germanic languages went through a similar process, but the names of polygons were calqued into native words: Ger. Dreieck, lit. ‘three-corner’, just as Old English, but as nouns, Viereck, Achteck, etc. Cf. Dutch _driehoek, Danish and Norwegian trekant (same meaning, different root), but Swedish triangel.

The Greek and Latin cardinal and ordinal numbers pop up in so many English and international words that it pays to learn them, sextuplets, quad bike, bicycle, tricycle, mononucleosis, and on and on.