Can I apply for a working visa while in Thailand on a tourist visa?

If this is a Thai company, they should be helping. If not, you need to be careful.

What is the job they've offered?

You have to go out of the country go to a Thai embassy and apply for a non b visa with the all necessary documentation.

Take fair warning from others' replies that it may be an uphill struggle for you if your potential employer is so completely clueless about employing a foreigner. The idea that they sent you to your own embassy to ask about working in Thailand is embarrassing.

Be aware that if applying for a visa at a Thai embassy or consulate in a neighbouring country, you will have to make an appointment online, you can't just show up. Some embassies get fully booked faster than others, and each has their own rules about booking appointments. Also be aware that in October some embassies have announced that they will be switching to e-visas. You'll have to go to the country the embassy is in and apply online from there. Then, the visa will be sent via e-mail and you will need to print it out to show at the airport and Immigration when you arrive back in Thailand.

Work visa, it does NOT exist in Thailand.

If you want to work for a Thai company you need 2 (TWO) different things.

to be allowed to work for that employer at that address doing as described as the job, you need a work permit from the Ministry of Labor.

To be allowed to be inside Thailand you need a visa, in this case the Non Immigrant B visa.

You can apply for a conversion of your tourist visa into this Non Immigrant B visa at your local Immigration Office, BUT as well as you and your employer need to meet the requirements.

As many times this seems to difficult for the employer, the easy way is to visit a Thai Consulate and apply for this Non Immigrant B visa. Your employer should contact that embassy to ask what documents are needed for this application.

Once you have the visa, you go back to Thailand get 90 days of stay, pick up your work permit and you can start working. 60 days after entry you can apply at your local Immigration office for the 1 year Extension of Stay based on working.

cost of travel, application fees, copies, hotel, food, drinks etc are depending on your preferences and you might ask your new employer for a donation.

The fact your potential employer told you to go to your own embassy suggests they don't know anything about applying for a Non-B visa. You can apply for one either at an immigration office inside Thailand or a Thai consulate outside Thailand. However in both cases you need paperwork from your future employer in order to apply, so your first step needs to be sitting down with them and seeing if they are able to provide such paperwork

I can't send you the graphic for applying in country for a Non-B visa (am on my phone) , but yes it can be done. A lot of employers prefer you to go abroad and get the visa, as it is less paperwork for them.