It’s certainly possible with a plugin. Do you have some SMS-sending service that you like? Do you have validated user phone numbers somewhere?
A plugin that blindly sent a text to whatever phone number that they put into a custom field is relatively easy (1-5 hours). A plugin that validated the phone numbers and gave the user means to opt out of future texts (rather than, say, sending a text when they got an email notification and having them use the email unsubscribe links) could quickly ballloon to 5-20 hours.
As suggested, a webhook that hit some external service could be done without a plugin, though you’d still need some means to acquire and validate phone numbers.