To answer the first part of your question: Yes you can make asynchronous requests with PHP, and even ignore the service's response. However, as you correctly say it's not a super great solution.
Asynchronous Requests
This excellent blog post on PHP Asynchronous Requests by Segment.io comes to several conclusions:
- You can open a socket and write to it, as described by this Stack Overflow Topic - However, it seems that this is actually blocking and fairly slow (300ms in their tests).
- You can write to a log file and then process it in another way (essentially a queue, like you describe) - However, this requires another process to read the log and process it. Using the file system can be slow, and shared files can cause all sorts of problems.
- You can fork a cURL request - However, this means you aren't waiting for a response, so if SendGrid (or some other service) responds with an error, you can't catch it and react.
Opinion Land 
We're now entering semi-opinion land, but queues as you describe (such as a mySQL one with a cron job, or a text file, or something else) tend to be very scalable as you can throw workers at the queue if you need it to process faster. These can be outside your user facing system (and therefor not share resources).
Queues
With a queue, you'd have a separate service that would be responsible for sending an email with SendGrid (e.g.). It would pull tasks off a queue (e.g. "send an email to Nick")and then execute on it.
There are several ways to implement queues that you can process.
- You can write your own - As you seem to want to stay on PHP/mySQL, if you do this you'll need to take into account a bunch of queueing problems and weird edge cases. However, you'll have absolute control and for a simple application maybe this will work.
- You can implement a self hosted task queue - Celery is meant to be a distributed task queue, øMQ (ZeroMQ) and RabbitMQ can also be used as Task Queues. These are meant to be fast and distributed and have had a lot of thought put into them. You'd need to benchmark them in your system to see if they speed it up. It'd also mean you have to host additional pieces yourself. This however, is likely to be the fastest solution from a communication standpoint.
- You can pass things off to a hosted task queue - IronMQ and Amazon SQS are both cool hosted solutions which means you wouldn't need to dedicate resources to them, additionally with IronWorkers (e.g.) you could have the other service taken care of. However, since you're trying to optimize a request to an external service, this probably isn't the solution in this scenario.
Queueing Emails
On the topic of queuing emails (specifically), this is something common to email senders. Like with everything else it means you can have better reliability (because if a service down the line fails you can keep it in the queue and retry).
With email however, there's some specific services out there for queueing messages. These are SMTP Servers. Theoretically you can setup a server like sendmail and then set SendGrid as your "smarthost" or relay and have the server send to SendGrid. It then queues and deals with service interruptions and sends mail with little additional code. However, SMTP servers are pains to deal with, even if they're just forwarding messages. Additionally, SMTP is even slower than HTTP to establish a connection and therefor probably not what you want, but it's good to know.
