I only see this tagged as a する verb in sankoku. N-grams are mostly 〜をする.

smk, shinsen, and meikyo refer to こつじき as 古風, 古めかしい, and 古い言い方, respectively. Daijirin has "古くは「こつじき」"

The kokugos have a second sense for こつじき that refers specifically to Buddhist monks-in-training asking for alms.

Google N-gram Corpus Counts
╭─ーーーーー─┬───────╮
│ 乞食をし　 │ 1,731 │
│ 乞食し　　 │ 1,097 │
├─ーーーーー─┼───────┤
│ 乞食をする │   574 │
│ 乞食する　 │   388 │
╰─ーーーーー─┴───────╯

@@ -17,0 +18 @@
+<re_inf>&ok;</re_inf>
@@ -26,0 +28 @@
+<pos>&vi;</pos>