Actually, it depends on what you really mean. If you're talking about to complete an application, then "заполнить заявление", it means to write there your name and other information. If you're talking about submitting an application to some institution, then "подать заявление".
And yes, the word "бланк" can be omitted.
But you can't use "заявлять" as "to complete" or "to apply", they have different meanings.