After researching the topic, I've determined that felt rather than carpet should be used. Below is a list of suppliers and their respective prices.

Option A

Vendor - www.dominoestoys.co.uk

Part No - GM21 GM001

Cost - £7.99

Description - 70x100cm piece of felt

=Option B

Vendor - www.onlinefabric.net

Part No - GLIGRE

Cost - $6.70

Description - 2 x 1 yards of felt

Vendor - www.onlinefabric.net

Part No - RAIPIR

Cost - $4.10

Description - 2 x 1 yards of felt

Vendor - www.ebay.com

Part No - 170266457478

Cost - $9.35

Description - 2 x 1 yards of felt

Vendor - www.save-on-crafts.com

Part No - 41212205090

Cost - $4.99

Description - 24 9.5 x 12 inch sheets

I'll try to find the exact thickness of all of the sheets, but the pictures looked like what we needed.