The card itself is apparently 40g–50g of CO2e.
http://www.icma.com/ArticleArchives/CarbonFootprint_SE2-12.pdf
You have to add in the paper and envelope and shipping costs, but that probably only adds 10–15g CO2e (I didn’t use the card figures from this source, as it was considering plastic-only, non-chip cards, but the source is the same consultancy as the previous link).
https://www.creditcards.com/credit-card-news/carbon-footprint-credit-card-1276.php
You would then have to add in the carbon footprint of sending a ~20g envelope by first class post. I’ve not been able to find figures for these.
However, I can’t imagine it is significant. I think the bigger issue of replacing a card is what happens to the defunct card. PVC is difficult to recycle and your old card almost certainly won’t be. It will take hundreds/thousands of years to biodegrade, and if incinerated lets of significant carbon (in relation to its size). So I think waste is probably the bigger issue when it comes to cards.
It feels like this is something that could be given some attention!