This is a common SaaS contract dispute and the answer depends on the specific language of your agreement. There are typically two documents in play: the Master Subscription Agreement (or Order Form) and the Terms of Service.
If your Order Form or MSA specifies a fixed price for a 12-month term, that is generally a binding commitment that the vendor cannot unilaterally modify. The ToS might govern feature changes, acceptable use, and general terms, but pricing in a signed order form usually controls over general ToS language.
The “we may modify at any time” clause is what is known as a unilateral modification clause. Courts have been increasingly skeptical of these, especially when they purport to allow material changes like a 40% price increase. In several jurisdictions, courts have held that such clauses are unconscionable or that they create an illusory contract (i.e., a contract where one party can change the terms at will is arguably not a real contract at all).
I would recommend: (1) review your Order Form and MSA separately from the ToS, (2) check whether the pricing language references the ToS or stands alone, and (3) send a formal written objection to the vendor stating that you do not accept the new pricing under your existing contract term. Do not just keep using the service silently, because the “continued use = acceptance” clause could work against you if you do not object.