Hi,
No, it doesn't require downtime. Just make sure the new switch is turned off before you connect the stack cable to the existing switches.
Follow this link to add a switch to your stack:
http://www.cisco.com/c/en/us/support/docs/switches/catalyst-3750-series-switches/71925-cat3750-create-switch-stks.html#stack3
http://www.cisco.com/c/en/us/td/docs/switches/lan/catalyst3850/software/release/3se/ha_stack_manager/configuration_guide/b_hastck_3se_3850_cg/b_hastck_3se_3850_cg_chapter_010.html#task_1177470
HTH