It depends how you define "sentence", which you haven't said.
Here is your question repeated with all the verbs highlighted in bold:
Does every sentence need a verb?
Can a sentence have no verbs in it?
If I say "I went for a run yesterday" then I am using a verb (to run) but if I say "This is a door" then am I using any verbs in that sentence? You can argue that 'is' is a verb because the verb 'to be' can be argued as 'to is' but that would be incorrect. An example where I can't see any verbs would be the first sentence of this page. 'Can a sentence have no verbs?' This sentence seems to have no verbs but still I am doubting if it doesn't have any verbs.
In short, does every sentence need a verb and if [it] doesn't, is it grammatically correct?
The verbs you used:
- to say: say (1st person, singular; present) ("I say" x 2)
- to go: went (1st person, singular; simple past) ("I went")
- to seem: seems (3rd person, singular; present) ("[it] seems to have")
- to run: run (infinitive) ("to run")
- to see: see (infinitive) ("I can't see")
- to have: have (infinitive) ("can [it] have" x 2, "[it] seems to have", "it doesn't have")
- to need: need (infinitive) ("does [it] need")
- to argue: argue (infinitive) ("you can argue"); argued (past participle) ("[it] can be argued")
- to use: using (present participle) ("I am using", "am I using")
- to doubt: doubting (present participle) ("I am doubting")
- to be is (3rd person, singular; present) ("This is a door", "[it] is a verb", "is it"); be (infinitive) ("to be", "[it] would be" x 2)
- to be: am (1st person, singular; present); be (infinitive) ("I am using", "am I using", "[it] can be argued", "I am doubting")
- to do: does (3rd person, singular; present) ("it doesn't" x 2, "does it need")
- can/could: can (present) ("can [it] have" x 2)
- will/would: would (subjunctive) ("[it] would be" x 2)
Note: modal verbs are defective in English because they have no infinitive, and there is no difference in the present tense conjugation.
As you can see, every sentence you used does contain at least one verb.
Your use of "run" is not a verb but a noun, because it is "a run". You can see that it's not a verb if you replace it with "meal": "I went for a meal yesterday", because "meal" is not a verb (except for a very old, rare usage).
The word "to" is not part of the verb. One of its many uses is to introduce a verb's infinitive. Your definition of "verb" seems to be a word that can take "to" in front of it. This is not the case. "is" is a conjugated form of "to be", and since it is not the infinitive, it can't take "to" in front of it.
"to be" can be used as a copulative verb (linking two things together), e.g. "This is a door" which can be thought of as "This = a door". The other use is as an auxiliary verb, a kind of "helper" verb. This allows expressing forms that we don't have a conjugation for in English, e.g. the progressive aspect which is used much more than the present tense, e.g. "I am using a verb" which can't be thought of as "I = using a verb".
Modal verbs are also truly verbs, but you may have missed them because you can't put "to" in front of them.
So, back to your original question. If we replace "sentence" with "grammatical utterance", then we can come up with many examples of grammatical utterance without verbs, some of which can be expanded to include a verb, some of which can't:
What? (What was that?, What did you say?, etc.)
What about this?
Most of these are interjections. There is a humorously written "story" which contains the following:
[...] if a sentence can refer or allude to itself, why not a lowly subordinate clause, perhaps this very clause? Or this sentence fragment? Or three words? Two words? One?
The purpose of the story is to explore the relationships between grammar, thought, and meaning, and most of the story would not be considered ungrammatical by most people (with a few very obvious exceptions). As such, "Or three words? Two words? One?" would be considered three distinct but related sentences without any verbs. There are other "sentences" that are deliberately ungrammatical as it explores the connection between grammar and comprehension. Most of these sentences use incomplete clauses, or just single words.
If you are interested, the full text is here: This is the title of this story, which is also found several times in the story itself. However, it is recommended that you have a very good understanding of English to be able to appreciate it fully.