So, I am designing a business card in illustrator and I want it to have a texture like this
Is this just the paper that it is printed on, or do I need to put some sort of texture in my design? Like do I just design it with a flat solid color?
This example that you show is 100% real paper. So you need to talk with printing house what paper you can use and prepare design for this paper.
You can print some paper texture on normal paper, but it will be not so realistic in my opinion and choosing other paper is better way :)
This is called Flexography, most of printing houses are able to do this, even the small ones, I'd dare to say that larger printing houses are 99% likely to work with it.
Usually it is not cheap, I'll take as example a budget printing house from my region (Porto Alegre, southern Brazil). 500 units of a 90x50mm coated business card colored on both sides costs about 70 BRL (~22 USD), if I put flexo in the quote, the price scales up to 300 BRL (~95 USD).
It gives an nifty and elegant look to your card, if you are willing to spend some extra bucks to stand out of the boring majority of business cards.
About how to do it: it is not that complex. First, you just must let the printing house know about what you want, and send them the template of the extruded area. Which basically consists in a black/white "mask" of the form of the bump, sepparated from the color artwork.
If you ever worked with UV Gloss Print / Spot Coating, it is the same workflow.
Before you promise your client a style of businesses card, ask around your local printers and get some prices as this technique might not be cost efficient, follow the link here for more about Letterpress description of letterpress process
To answer your question:
Yes, you can use a texture over the solid color in your design if you are printing on textured paper to achieve this look, but it will depend on what paper you are using. Like the others have said, talk to your printer first and the words you want to use are "letterpress" and "embossing."
Check out Moo's sample for this style: https://www.moo.com/us/products/letterpress-business-cards.html