Is "anxietizing" really not a word?
Solution 1:
As always, the question is, what do you mean by "a word"? Since -ize is a productive verb-forming suffix, you can attach it to anxiety to produce a word that most English speakers would be able to decipher.
As a survey of major dictionaries will show, however, anxietize is not a standard word, and its lack of an OED entry suggests it has never been. To describe something as creating anxiety or for making someone anxious, one would use anxiety-producing, anxiety-inducing, anxiety-generating, anxiety-triggering, or a similar compounds, likewise similar terms like worry, dread, panic, doubt, or unease.
Alternatively, you can use a participle for verbs of inducing concern: I find sharks unnerving, dismaying, concerning, or distressing. Popular in slang hyperbole at the moment is triggering, in reference to trauma triggers.
Solution 2:
Anxietize (and depress1) are reflexive verb forms of anxiety (and depression) coined by Albert Ellis, and used in some medical circles to give the patient the perspective that these are not wholly things which happen to them, immutable, but something over which they have some degree of control or agency, a key tenet of CBT.
They have little to no currency outside of this limited context.
e.g.
This terminology shows that I now depress and anxietize myself — but that I need not do so in the future. It uses verbs and adjectives to describe my (and others') distressing, and avoids the hazards of always using negative diagnostic labels - such as depression and anxiety - that overgeneralize and impply that I am unable to change my thinking, feeling, and behaving.
- Overcoming Resistance: A Rational Emotive Behavior Therapy, Albert Ellis
I noted in the first edition of Reason and Emotion in Psychotherapy that people who anxietize, and particularly those who panic, frequently strongly create...
- Overcoming Destructive Beliefs, Feelings, and Behaviors, Albert Ellis
1. Of course the verb to depress long predates this specific reflexive medical usage "to depress oneself" and is widely used.
Solution 3:
A more appropriate adjective meaning "making anxious" is anxiogenic (with the antonym anxiolytic).
However these terms are only commonly used in scientific/medical contexts rather than in common speech, and almost exclusively reserved for psychoactive substances rather than the subjects of specific phobias; for common usage you are limited to the hyphenated words suggested by @choster.
Solution 4:
Well... anxieting, kinda exists.
Any noun (or occasionally other parts of speech) in the English language can be "verbed".
Glasser suggests that to be consistent with the notion of total behavior and choice, we need to say "I'm anxieting," or "I'm depressing." To summarize the concept of total behavior, consider the example of an assistant professor waiting to hear about his application for tenure. He is sitting at home (action) "anxieting" (feelings), wishing his chairperson would phone the results ...
Texas Tech Journal of Education (1986)
The term is perhaps not wholly conventional, but native speakers have always been discovering ways to overcome language restrictions, which is why English is so flexible in the first place.
Alternatively, the OP could use any one of the following present participles in their sentence;
alarming, perturbing, upsetting, unnerving, unsettling, disconcerting, worrisome, or simply terrifying.