I have been reading the manga series Vagabond by Takehiko Inoue which is set in the late Sengoku Period of Japanese History. One of the defining moments for the characters centers on the Battle of Sekigahara and its aftermath. In one instance in Chapter 164 shows that Tokugawa Ieyasu put a bounty on Ishida Mitsunari. Is there evidence of this specific instance? Are there other instances of this happening during the Sengoku Period?
Yes, that was indeed a thing. Although it almost always varies with circumstances.
The case of Sekigahara was clear - Mitsunari (and other Western army commanders) were the "traitors" - they abused power under Hideyori (as Ieyasu had spun the narrative, which isn't entirely incorrect) and broke the peace Hideyoshi fostered for so long. They were legally made enemies of the Toyotomi political system (which still was the central system Japan was running with, although Ieyasu's actions would later slowly dismantle its influence), and had to be chased down and punished. A lot of the Western army daimyos commited suicide or attempted to evade capture. Ukita Hideie, Ankokuji Ekei, Shimazu Yoshihiro, Konishi Yukinaga all ran; while people like Onogi Shigekatsu found themselves in a situation where suicide was the kindest outcome. The bounties were big, too. Tanaka Yoshimasa's whole career (more than doubling his amount of land) was because his men found Ishida Mitsunari. Otherwise, Yoshimasa played minimal role in the key battle and would've gotten minimum rewards.
A similar thing happened after the siege of Osaka - where a lot of people who joined Toyotomi Hideyori's side ran after the castle fell. Chosokabe Morichika and Akashi Teruzumi were good examples, and bounties were placed on them.
Another example would be Oda Nobunaga hunting down ex-Takeda retainers after Koshu conquest. In this case, the bounties were made to ensure the ex-Takeda retainers (or even Takeda family members) couldn't rally up local support and form obstacles against the new Oda governance. Peasants flooded the meat market (metaphorically) with heads of their previous lords for a quick buck (good payout, too).
In any of the above cases, the need for these bounties were clear: to cleanse off any remaining threats. Mitsunari and his fella co-conspirators were threats to Ieyasu (and as he claimed, the Toyotomi regime), and so were the later Morichika and Teruzumi to the Edo bakfu. Ex-Takeda retainers were threats to the rule of the new lords Nobunaga placed in territories previously held by the Takeda, and we do indeed see their power later (when these men started rebellions and began pushing off or even killing Oda lords).