Why have it be by text? I would be wary if I were the customer. The last thing I want is to find myself getting marketing calls or texts because the only way to sign up for the warranty is by text. I may return the product if I was forced to choose between forgoing the warranty and giving my cellphone number out to a company that I don’t know will use it only for the intended purpose of providing warranty service.

If by “customer engagement” you mean that you intend to use the numbers of the customers that register for future marketing, that is absolutely not allowed. If it is purely for warranty purposes…not technically a violation (though Amazon still may interpret it as diverting customers off of Amazon), but not the best choice for warranty registration either.

No. Those are two Prohibited Seller Activities and Actions.

Attempts to divert transactions or buyers:

Any attempt to circumvent the established Amazon sales process or to divert Amazon users to another website or sales process is prohibited. Specifically, any advertisements, marketing messages (special offers) or “calls to action” that lead, prompt, or encourage Amazon users to leave the Amazon website are prohibited. This may include the use of email or the inclusion of hyperlinks, URLs, or web addresses within any seller-generated confirmation email messages or any product/listing description fields.

Inappropriate email communications:

Unsolicited emails to Amazon customers (other than as necessary for order fulfillment and related customer service) and emails related to marketing communications of any kind are prohibited.

https://sellercentral.amazon.com/gp/help/200386250

Funky

As a person who does not text I’d be very unhappy if my only form of registering my warranty were through that process.

The OP’s question is not whether you personally would like to register for a warranty via text. They are asking if it violates Amazon’s rules of engagement and the answer is NO. It does not.

@FunkyMonkey cited two prohibited seller activities. The first one ONLY pertains to diverting traffic away from Amazon’s sales process. The OP’s sticker does not violate this. This rule is in place to prevent sellers from circumventing paying Amazon its dues. This rule is not violated by asking customer’s to register for an extended warranty through text or through their company website.

The second rule is obviously regarding email correspondence. Again the OP’s strategy does not violate this rule.

It would be a better strategy to have that information printed on the packaging of your product, or with the included manual.
It’s still diverting buyers, even if its not for another sales channel. So it’s a thin line between this being okay and a suspension.

I just ordered a bunch of Cuisinart items. Every single item had a lifetime warranty postcard enclosed to register via text, email, or going to their site. All of it Amazon fulfilled.