Bards can scribe scrolls if they are proficient in Arcana

Bards are spellcasters

With time and patience, a spellcaster can transfer a spell to a scroll, creating a spell scroll.

The DMG outlines what it means to be a spellcaster:

If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.

Even a level 1 bard can cast several spells, so they qualify as spellcasters.

The bard must be proficient in arcana and know the spell they want to scribe

Xanathar's Guide to Everything lays out the (optional) rules to scribe a scroll as follows:

[T]he character must have proficiency in the Arcana skill [...] Moreover, the character must have the spell prepared, or it must be among the character’s known spells, in order to scribe a scroll of that spell.

Beyond this, there are no requirements. Therefore, any level of Bard can meet the requirements as long as they have proficiency in Arcana and if they know the spell they are attempting to scribe.

Your DM must allow magical item creation and downtime must be used

A spell scroll is a magic item and thus falls under the rules for creating magic items. The rules for creating magical items are optional and must be allowed by your DM before they can be used. So talk to them.

Also note that the rules for scroll scribing in XGtE specifies that scroll scribing is a downtime activity. So they would use the downtime rules. Consult your DM to see how they want that to work as well.

Jeremy Crawford has ruled in as well:

Q: Can Bards make scrolls?

A: Using the downtime rules, any spellcaster can create a spell scroll. See the "Dungeon Master's Guide" ("Crafting a Magic Item," p. 128) for more information. "Xanathar's Guide to Everything" ("Scribing a Spell Scroll," p. 133) contains an alternative method.