Basically, three inter-related questions:
Could you weave a seamless garment in the 1st century? I know you can knit one, but that technology wasn't invented yet. Or how else would one be constructed?
Is there historical evidence of seamless garments dating back to this period? I don't mean supposed holy relics discovered hundreds of years later.
Why does the author of John mention this detail? Is there a tie-in to Judaism? Is this a trope common to other mythologies of the time, as is the virgin birth?
The text you have in mind is John 19:23-24. Here's my translation:
Then the soldiers, when they crucified Jesus, took his garments and made four portions, a share for each soldier, and the tunic. Now the tunic was seamless, woven from the top as a whole. 24 Then they said to one another, ‘Let’s not tear it, but cast lots concerning whose it will be’; so that the writing might be fulfilled [which says], "They divided my clothing among them, and they threw a lot for my garment." So the soldiers did these things.
So, there are a few things to recognise first. The garment in question was a kind of tunic that was worn under an outer cloak. Josephus in Antiquities 3.161 records the same concerning the high priest’s robe. Ancient commentators frequently found spiritual significance in the seamless nature of the robe, but that does not appear to be John’s point, which is instead centred on the dual-fulfilment of how he understands the quotation from Psalm 22:18. Theodore of Mopsuestia provides a helpful comment in that he notes that such methods of weaving were common in the time of Christ. So, in answer to your first question, a seamless garment was not out of their technological reach, and would have been possible. John’s point seems to centre on, rather, that (a) the clothes were divided, and the tunic was left-over, so that (b) they could also gamble for the clothing, thus fulfilling both clauses in the psalm quotation.
The tie-in to the high-priest does not appear to be John’s point, nor does ‘seamless garments’ have much other resonance in Judaism, certainly not as a common ‘virgin-birth trope’.
The High Priest in the Temple wore a seamless robe as one of the sacred garments. Jesus wearing such a robe seems to make some kind of identification (either historical or editorial) of Jesus as being the true High Priest. Either Jesus was making a theological statement by wearing such a robe, or John was making his own theological statement by putting Jesus in one. It's not historically impossible that he really wore such a garment. Others did.