Unless my googling skills have let me down, it seems unlikely that the resort itself runs any excursions. To have them available to you, you'd need to book throuh a company which provides coach holidays (such as this one
https://www.justgoholidays.com/itineraries/8238-hayling-island---warner-entertainment-break ), with that coach company then offering the exursions. Warner will assume that most of their visitors will either be using such companies or will have their own cars.
For a definitive answer though, phone the site on 023 9246 3976.
I know you've asked elsewhere tonight, but it's only about a 4 minute walk to the bus stop to catch the service 30 bus into Havant. The timetable is here:
https://tiscon-maps-stagecoachbus.s3.amazonaws.com/Timetables/South/Portsmouth/Portsmouth%20Route%2030%2631_23.07.17_webfile%201.pdf
(Use the times for Mengham Corner, which is the closest point shown on the timetable. The bus ride takes about 20 minutes).
It only takes between 10 and 20 minutes by train from Havant to Chichester, with 4 trains per hour. The off-peak return fare is £5.00 (or £3.30 with a railcard).
There are typically 3 trains per hour to Winchester from Havant, with journey times between 50 mins and 70 mins. The off-peak return fare is £15.90 (or £10.50 with a railcard).