You simply need a way to send SMS messages that allows you to send a message where you can specify the sender.
That's certainly the easiest way to do it, but having that level of access is tough. My company provides apps and APIs to allow people to send/receive SMS. We do a lot of work to ensure that the "sender" you set is a phone number belonging to you.
We connect directly to SMS aggregators, and they generally don't do much validation on the sender phone number that our platform provides. For example, I can easily send myself a message from a fake number like 15551234567, a toll-free number, someone's mobile or landline, etc.
That said, carriers and aggregators constantly monitor for spam and other odd usage patterns. They will block phone numbers and/or originators of bad traffic. An occasional spoofed message or two could fly under the radar, but it's in the interests of any entities that grant you access to the SMS world to prevent you from sending these types of messages.
tl;dr: Spoofing is possible, but you'll have a hard time finding anyone willing to give you that access, since granting that access could jeopardize their entire business.