Given that you're comfortable with a shared PHP host, I'd assume this would comfortably run on a single Heroku dyno?
If so, a single dyno running 24/7 is $7 a month. If that's still too expensive, you could use a free dyno and setup an AWS Lambda which triggers it every ~15 minutes, which would effectively be free.
There's also a number of niche hosts, e.g. zeit[0] (FWIW, haven't used it before, not necessarily endorsing or recommending them) which offer free or cheap managed hosting.
If you're interested in a solution which is potentially a little more interesting and also effectively free for low traffic, you could run your PHP application within AWS Lambda[1].
Unfortunately I can't comment on legit shared hosts (e.g. Bluehost, Godaddy, etc.) because I generally try to avoid them.
Edit: You could also put it on AWS Lightsail, Linode, Digital Ocean, etc. but from your mention of Heroku and desire for shared hosting, I'm guessing you don't want to configure a VPS.
[0]: zeit.co
[1]: https://akrabat.com/serverless-php-on-aws-lamda/