No it's not secure.
The government and your mobile carrier can easily read it(so forget about sending sensitive information or not so legal SMS).
While i find it difficult to SMS be intercept while in traffic remember that both you and the receiver will have this SMS until you deleted it and this introduce some risks like you having your phone stolen and don't use password on it or the receiver handing over his phone to a third person and even forward it to more people.
Now that i have answered the SMS part i must say about the bigger picture you are not seeing:
How do you know if the owner of the store is really who they say they are? How do you know this store is even real? I find pretty weird that you are communicating with them via email and even more weird they say they don't support credit card payment via some website...even more weird if their sites don't use https:// at all.
Not supporting https:// is a red flag at first sight a certificate is pretty cheap nowadays and if they want to accept credit cards without even caring to use TLS shows they lack good security practices...
They could in theory accept credit card payments without using TLS if they don't host the checkout or store credit card details in any way and there are many third party payment processors like PayPal or Stripe, and most banks provide some ready-to-use solutions for e-commerce but not using https:// still a red flag.
This really seems like a scam to me and you should never by any way SMS your credit card info to anyone no decent and reputable store would ask you to do it.