Is a text message legally binding?
Yes, but the terms of the message need to be clear enough to ascertain the parties' intent at the formation of that contract or agreement.
A contract does not even need to be in writing. There are also oral contracts and implied contracts, the latter referring to contracts which are inferred from the parties' conduct. A contract such as the agreement you describe here is binding regardless of its form. It is just easier to prove the existence of a contract if it is in writing.
You did not specify your jurisdiction. If it is in the US, the price tag --rather than the downpayment-- of the object of the contract (i.e., the puppy you intend to buy) determines whether your complaint would need to be filed in Small Claims court. Generally speaking, parties to a dispute in Small Claims court have to represent themselves.
Two remarks are pertinent. First, developing writing skills is utmost important not only for litigating a dispute, but also during the process of formulating the terms and conditions of a contract/agreement. Your post indicates that you seriously need to work on that.
Second, the end of your post reflects that one of your managers violated labor law(s), which to most of us would be more worrisome than the controversy about the puppy. Legislation in most or all jurisdictions outlaws the act of withholding an employee's compensation regardless of its form (salary, commissions, and so forth). You might want to gain acquaintance with the labor laws of your jurisdiction so you can assess whether or how to proceed (does legislation require the employee to "exhaust administrative remedies" prior to filing in court? are administrative remedies optional? do these exist at all?), even if only to ascertain whether the deadline for filing the corresponding claim has elapsed.