I was thinking of what role the Order of Arkay in High Rock might play outside the rural south west (Ilessan Hills and Shalgora) where it's more popular and all I could think of was burial rites. Arkay is the god of birth too so if there are rites for dead, are there rites associated with birth (in the Imperialised Tamriel in general)?

The burial rites prevent necromancers from raising corpses. This is like turing the wheel of life backwards, so maybe the birth equivalents would be infant mortality and stillbirth. So would they use blessings to prevent that? The easy and probably boring way to do this is to rip off baptism. I thought that children might be protected from daedric abduction like how fairies were deterred from stealing baptised children, but that anti-daedra protection sounds more up the Akatosh Chantry's street in my headcanon.

Interesting idea.

Why not, I could easily see a priest of Arkay bless an expectant mother.

Sounds well enough, preserve life, along with death, and keep that cycle going. Help expected mothers with the birthing process, along with rites and sacraments for when the child is born, such as cleaning it, making sure it's not stillborn, and also what you put, no daedric abductions and whatnot.