I am a service provider with money in my Rover account from services provided in 2018. If i leave the money with Rover until 2019, will I incur a tax liability this year or next? In other words will I get a 2018 1099 that shows the payment this year even if I leave the money in my account?
First of all you probably won't get a 1099 but you don't need it to report your income on Schedule C.
Yes, you have to report the income when it is earned. It doesn't make any difference if you leave it in a noninterest-bearing account at Rover.
All income has to be reported in the year that it is earned if you are using the cash method of accounting. It doesn't matter when you withdraw the money from Rover, you earned it in 2018 therefore the income should be reported on your 2018 tax return. You should use Schedule C for reporting your dog sitting income and expenses.