No, you cannot work on a visit visa. It is illegal. In fact, you should report this situation to the nearest UAE embassy as there is a high chance this employer is also violating the labor law by trying to skirt the visa process.
If you do attempt to enter the country, there is a good chance you will be denied and then deported.
There are also other things wrong with this situation:
An offer letter is not a contract of employment. The contract is a legal document, based on which you have certain rights, and is registered with the UAE government.
The fact that the company is not mentioned as a sponsor tells you that the job opportunity is not genuine.
Normally with companies like this, the offer letter states a very high salary, and if you manage to enter the country you are put on a different salary (usually paid in cash) and the job you are doing is not what is mentioned in the offer letter. You also are not given benefits you are entitled to by law (like mandated days off, tickets to your home country, insurance, etc.) because you are an illegal worker.
Do not accept this offer
As far as the visa situation, in UAE, there are multiple kinds of short visit visas:
- Transit visas, valid for 72 hours.
- On-arrival visas, valid for 30 days; available for certain nationalities only.
- Electronic visas, you have to apply before you enter the country. 30 days, available for residents of the GCC and some others.
- Tourism visas. Available for 30, 60, 90 days; and some available with multiple entry.
None of these visas allow you to work. It states so directly on the visa form.