Bright Horizons Back-Up Care
My kids go to daycare & school in Oakland & I work in San Francisco. I get those calls. I think most daycare places get that it's going to take you some time to physically get from point A to point B and should reasonably be willing to keep your kid there as long as you are communicative & in transit - even if, for example, in my case it's probably going to take 45 minutes to get to them. It is what it is. I even have family in the East Bay, but it would still take them probably 45 minutes to get the kids too so that doesn't really make a difference.
My other thought would be: do you have any close neighbors (presumably also by your daycare) that are maybe retired and would be willing to pick them up and watch them temporarily if you gave them a key? Is your daycare walking distance so you wouldn't have to worry about car seats in cars? Even that wouldn't be a guarantee they would be around, but may offer peace of mind. Of course, if your kid was SUPER sick and contagious the neighbor probably wouldn't want to be exposed...
If your question is more: how can I get someone to watch my child when they're sick so I can still go to work? a) Does your employer offer Bright Horizons back-up care as a benefit? It is great, they will send someone to your house with fairly short notice for a reasonable fee, even if your kid has a "mild" fever (they won't administer medicine though, so it has to be pretty mild). I believe your employer has to offer this though I don't think you as a consumer can just randomly use it.... my work didn't, but I suggested it, and we now have it (it took about a year to get them to go through all the hoops though). b) Otherwise, you just have to call in sick and/or leave mid day. Stressful at the time, & in the moment feels like a big deal - but most people get it. It probably feels like a much bigger deal to you than it does to your boss or anyone around you.