Depends on how the question is structured. If it's open enough for a mental reservation, you could do so. Such as if the question were "Have you had the vaccine?" You could respond, "I've had it." as in saying "I've had it with this vaccine business." Another suggested by someone else on here was saying you are fully up to date with your vaccinations, without specifying which.

But mental reservations walk a fine line between truth and lies, so it really can only be applied in particularly strenuous situations and rarely. Otherwise you could fall into a habit of lying. At least, this is the advice Father gave us in his sermon recently.

Statement of Declination for Offer of Influenza Vaccine Product 1. YOUR NAME, a living woman retain and reserve all of my God-given rights including sole possession and sole use of all my biological materials which are granted to me by my Creator. My employer and location of employment have offered an influenza vaccination product to me pursuant to and satisfying the requirements of YOUR STATE Health & Safety Code COPY AND PASTE INFLUENZA HEALTH AND SAFETY CODE HERE. I retain the right to decline all attempts to access, influence and or otherwise alter any and all of my God given biological material and or biological systems which are unique, flawless and original design and craftsmanship of my Creator and of which my Creator has granted me sole possession, proprietorship and use of. I require that any and all product offered to me by my employer or workplace be both entirely retrievable from and also removable in its entirety from my body, person, and womanhood at the conclusion of each and every work period and or work shift and also and again at the completion of my contractual obligations with my location or employment, and or employer. Pursuant to my above statement, I decline the offer for influenza vaccination product. By: YOUR SIGNATURE DATE YOUR PRINTED NAME All Rights Reserved

Adjust as needed. There is a health and safety code for every state you can look up.
Legally it is important to address the vaccine as PRODUCT that is being OFFERED to you, which gives you right of refusal. Then the ball is no longer in your court. 
Also the last paragraph  which states that the product has to be eradicated from your body when not at work actually shows a form of cooperation by you that the employer cannot comply with.
Believe it or not  I have read this declination has worked in the past  and somehow ticks a legal-ese box that flummoxes  employer legal departments. Usually they are unable to respond to this and will leave you alone. I personally have not had the need to use this myself.
  

Have you taken any testing before?  I'm assuming it's the nasal kind?

I agree.  Don’t take the nasal swab tests.  They puncture your nasal/brain cavity and will screw up your sinuses (maybe worse) for life.  
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Time to look for another job.  

Ask the employer how much does the company get for testing for covid?

And if you get covid after getting tested or vax, can you hold them responsible for any problems?

I heard from a manager that our boss is planning to ask all of us tomorrow at a team building event what our vaccine status is. Wondering how I should answer.

Tell him you identify as being vaccinated.  

You can also request a medical exemption.  If the mandatory issue comes my way, I will apply for both, medical and religious.