Learn Hardcore Japanese: ryousyuusyo wo kudasai. kaisya ni teisyutusimasu. - Please give me a receipt. I’ll submit it to the company.

ください functions like a verb meaning “please give (me)” here, so the thing you’re asking for is marked as a direct object with を. Pattern: Noun + を + ください (e.g., 水をください).
When you’re asking someone to do an action, use the verb in -て form + ください (e.g., もう一度言ってください).

What’s the difference between 領収書 and レシート?

領収書: A formal receipt often used for reimbursements/expense reports. It can include a recipient name (宛名), purpose (但し書き), and a stamp.
レシート: The standard printed till receipt; itemized and time-stamped. Many companies accept it, but some still require 領収書.
You may also see 領収証 (alternate form; similar meaning). For billing before payment, that’s 請求書 (invoice), not a receipt.

Is 領収書をください natural and polite enough at a store?

Yes, it’s perfectly natural and polite. Slightly softer or more formal alternatives:

領収書をお願いします。
領収書をいただけますか。
領収書を発行していただけますか。 (more formal)

Can I drop を and just say 領収書ください?

In casual speech, yes: 領収書ください is very common at the register. In careful or formal speech, keep を: 領収書をください.

Why is に used with 提出します in 会社に提出します?

に marks the recipient/target with verbs of giving/sending/handing in. With 提出する, the natural target marker is に: Xを Yに提出する.
へ emphasizes direction and is uncommon with 提出; 会社へ提出します is possible but sounds less natural than 会社に提出します.

Does 提出します here mean future, even though it looks like present tense?

Yes. The polite non-past (〜します) covers both present and future in Japanese. Context supplies the time nuance, so 提出します can mean “I will submit.”

The second sentence sounds like a reason. How can I connect it more smoothly?

Use a reason connector:

会社に提出するので、領収書をください。 (soft, polite)
会社に提出しますので、領収書をお願いします。 (more formal)
会社に提出するんで、領収書お願いします。 (casual) You can also keep two sentences in speech: 領収書をください。会社に提出するので。

Does 会社 here mean “my company”?

In this context, yes—会社 typically implies your own employer. If you want to be explicit, you can say うちの会社に提出します. If you meant some other company, you’d specify its name.

Should ください be written as 下さい?

Modern style prefers ください (hiragana) when making requests (after nouns or verb-て). You’ll still see 下さい on signs, but writing ください is the safe, standard choice for learners.

Is 提出する transitive? What’s the omitted object in 会社に提出します?

Yes, 提出する is transitive. The object (here, the receipt) is simply omitted because it’s obvious from context. Full form: 領収書を会社に提出します.

What’s the difference between 提出する and 出す?

提出する: Formal “to submit/turn in” to an authority (company, school, office).
出す: General “to put out/hand in/send.” In casual contexts, 会社に出します can work, but for official paperwork, 提出する is the better fit.

Are spaces normal in Japanese writing?

No. Spaces were added for teaching clarity. The standard way is: 領収書をください。会社に提出します。

How do I ask them to write my company name or purpose on the receipt?

Useful phrases:

宛名は〇〇株式会社でお願いします。 (Please make the recipient name “〇〇 Co., Ltd.”)
宛名は会社名でお願いします。 (Recipient name: my company, please.)
但し書きは「〇〇代」でお願いします。 (For the purpose line, please write “〇〇 fee/charges.”)
If you need a date: 日付は本日でお願いします。