The Pass Wide and Slow Wales campaign has collected 1,300 signatures on the assembly's e-petition website.
It wants an annual road safety awareness campaign explaining to motorists how to react around horses.
The British Horse Society found that since 2010 there have been 2,000 road accidents in the UK, with 1,500 because of cars passing too closely.
As a result of these, 180 horses and 36 riders have died.
Awareness rides were planned for Penarth, Vale of Glamorgan, Swansea, Neyland in Pembrokeshire, Machynlleth, Powys, Flintshire and Porthmadog in Gwynedd.
Any petition with over 50 signatures is considered by a committee to be debated in the assembly.

Councillors are unhappy Ordnance Survey (OS) maps only refer to Black Rock Sands near Porthmadog by that name.
But it is also know locally as Traeth Morfa Bychan - after the village next to the shoreline.
OS told the authority the name was adopted after consultation with the council, and they had been told the English version was most commonly used.
"During the council's Language Committee, members declared their commitment to protecting indigenous Welsh names and places in Gwynedd," said a council spokesperson.
They said they would continue to collaborate with agencies such as OS to "ensure that the correct names are used wherever possible".
"In this case, the discussion with the OS will continue regarding the possibilities for changing the name that appears on their maps," added the Gwynedd official.
"In the meantime Gwynedd council will consult with community councils in the specific area, to gather evidence that will enable the council to submit robust evidence to the OS for changing the name that appears on their maps."

19 April 2017 Last updated at 17:21 BST
So who's in and what's it all about? Here's the lowdown...
Awareness rides are taking place to try and cut the number of people on horseback injured or killed on roads.
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