Which software can be used to estimate a non-recursive SVAR? Not sure though if this question belongs here but I found nothing so far on google. Or researchers using them do it manually?
EViews can do SVARs with custom restrictions, including some presets. You may check out their online documentation. Btw, this one is the only one to my knowledge that also does long-term restrictions on the residuals.
Pretty sure that JMulti can also do the kind of restrictions you are after. You can get it here for free. It's quite old so I am not sure about system requirements.
Statsmodels (via python) might be able to do it too, but only with presets.
R has it as well.
Stata, but unsure about custom restrictions.
Note that there are number conditions for implementing restrictions on contemporaneous relationships between variables. The EViews documentation explains that. From your comment it looks like you may have under-identified the A matrix (i.e. asking for too many coefficients to be estimated). That might be the reason why you get an error.