For starters, we're submitting a petition addressed to the mayor and running a signature campaign.
Literal
First, mayor [addressed-to-あてに], petition [obj-を] submit-[and-て], signature-movement [obj-を] are-doing.
あてに (宛てに) is the postposition 'addressed to,' extended from mailing into the figurative sense of directing a document at a specific recipient. The て-form of 提出する chains into the second clause, the standard conjunctive for actions sharing a subject. The final ~ている here is the progressive 'currently running' sense — the campaign is an ongoing activity in progress.