Can I use trisodium phosphate for killing weeds?

No.
TSP is best used as a cleaning agent, like when you want to paint your house but need to clean it first.

I’ll admit that I’m not familiar with TSP; however, my research leads me to believe that it would possibly kill more than the weeds. I usually use Distilled Vinegar to kill my weeds especially around my Azalea. I make a 50/50 blend of of the Vinegar heated then pour over the weeds. In my case, I ge the added benefit as my Azalea loves the extra acid from the Vinegar.& the weeds don’t!!!

Boiling water generally kills weeds extremely effectively, but that also depends on how hearty the plant is and how deeply rooted it is. Ditto for diluted vinegar. Also, diluted vinegar won’t be as effective if the ground is already damp.

Keep in mind that this is a chemical with serious potential to cause physical harm. Do your research! Don’t “experiment” with it. Learn how to use it.

Years ago I tried to buy some to clean a wall before painting, and discovered that it was illegal to even purchase in my state.

I tried to find use for this chemical as a pesticide and could find no instructions. That troubles me. Considering what I’m looking at in the link below, I am alarmed at your cavalier attitude about use of this chemical. Check this out: http://www.pesticideinfo.org/Detail_Chemical.jsp?Rec_Id=PC34419

I also found instructions for a related chemical as a bactericide. https://www3.epa.gov/pesticides/chem_search/ppls/000148-00676-19680219.pdf

You know what else kills weeds? Motor oil! I don’t recommend that you do this, but this is to tell you what happened.
I left a pan of motor oil that I had drained from my car out on the ground in my yard, and forgot about it. When it rained, the pan overflowed and the oil got spread out on the ground. Nothing grew on that patch of ground for years. I was amazed.

Also, TSP is most often used to prep a wall for painting. Be careful, and read ALL the instructions.