If you go to Hamad Hospital vaccines for kids under 1 are free. You can go to the private hospitals but they are much more expensive unless you have insurance that cover them. Both my kids were completely vaccinated in Doha (born here, now 1 and 3). The only concern might be transferring the records to your home country when you return. Each country has different vaccine schedules. But, if you are here for a while you might as well do the Doha schedule and finish it all here. For example, in the US they don't do BCGs, but since I figured my kids would be here the first few years of their lives so it was probably a good idea to do it.

I'm pretty sure you don't need all the vaccines before arriving in Doha. Check with your employer, but I can't imagine they will turn away your 3 month old w/o vaccines since I know they don't check vaccines for adults at all!