wǒ xiǎng bǔbàn yínhángkǎ.

Here, 想 means want to or would like to, not think.

In Mandarin, 想 can have more than one meaning:

• 想 + verb = want to / would like to
  • 我想补办银行卡。 = I want to reissue my bank card.
• 想 + clause can mean think
  • 我想他今天不会来。 = I think he won’t come today.

So in this sentence, because 想 is followed by the verb 补办, it expresses desire/intention.

补办 means to reissue, to replace by going through the formal process again, or to apply for a replacement.

It is often used for official documents or cards, especially when something is:

• lost
• damaged
• no longer usable

For example:

• 补办银行卡 = reissue a bank card
• 补办身份证 = replace/reissue an ID card
• 补办手机卡 = replace a SIM card

It is more specific than just 办, which simply means to handle, to do, or to apply for.

银行卡 is a fixed compound noun meaning bank card.

In Chinese, many common noun combinations are formed directly without 的:

• 银行卡 = bank card
• 信用卡 = credit card
• 手机卡 = phone card / SIM card

Using 银行的卡 would sound more literal, like the card belonging to the bank, and is not the normal everyday way to say bank card.

So for the standard item, 银行卡 is the natural form.

Not necessarily.

我想补办银行卡。 is a natural sentence, especially when the context already makes it clear that you mean my bank card or a bank card.

But you can also say:

• 我想补办一张银行卡。

Here, 张 is the correct measure word for cards.

The difference is:

• 补办银行卡 = reissue a bank card / my bank card
• 补办一张银行卡 = reissue a bank card, with the object stated more explicitly as one card

In real life, both are possible, but without 一张 is very common.

The word order follows a very common Mandarin pattern:

subject + modal/auxiliary verb + main verb + object

So here:

• 我 = I
• 想 = want to
• 补办 = reissue / replace
• 银行卡 = bank card

So:

• 我想补办银行卡。
• literally: I want to reissue bank card

This is normal Chinese sentence structure.
A verb like 想 comes before the main action verb.

Yes, but the tone changes a little.

• 我想补办银行卡。 = I’d like to reissue my bank card.
• 我要补办银行卡。 = I want/need to reissue my bank card.

想 usually sounds a bit softer and more polite.
要 can sound more direct, and depending on context, sometimes stronger.

At a bank counter, 我想补办银行卡 is usually the safer and more natural choice.

It is pronounced:

• 银 yín
• 行 háng
• 卡 kǎ

So: yínhángkǎ

The character 行 has multiple pronunciations in Mandarin. Two very common ones are:

• xíng = okay, to be possible
• háng = profession, row, line; also used in 银行 for bank

So:

• 银行 = yínháng, not yínxíng

This is something learners simply have to memorize as part of the word.

Not only for a lost card.

补办银行卡 can be used when the card needs to be replaced or reissued, for example if it is:

• lost
• stolen
• damaged
• demagnetized or no longer working

If you want to be more specific, you can add the reason:

• 我的银行卡丢了，我想补办。
  My bank card is lost, and I want to get a replacement.
• 银行卡坏了，我想补办一张。
  My bank card is broken, and I want to replace it.

So the sentence by itself is broad enough for several replacement situations.

Yes. 我想补办银行卡。 is grammatically complete.

Chinese often leaves out information that is obvious from context. In English, you might expect something like:

• I want to replace my bank card
• I want to apply for a replacement bank card

In Chinese, if the situation is clear, you do not always need to specify:

• whose card it is
• why it needs replacing
• whether it was lost or damaged

That is why the short sentence still sounds natural and complete.

A few natural options are:

• 您好，我想补办银行卡。
  Hello, I’d like to reissue my bank card.
• 您好，我想补办一下银行卡。
  Hello, I’d like to get my bank card reissued.
• 你好，我想补办一张银行卡。
  Hello, I’d like to replace a bank card.

A couple of notes:

• 您好 is more polite than 你好
• 一下 often softens the tone a little in conversation
• In real life, staff may then ask:
  • 是挂失补办吗？ = Is this a replacement after reporting it lost?
  • 身份证带了吗？ = Did you bring your ID?

So your sentence is already good, and adding 您好 makes it even more natural in a service situation.