Learn Russian: Вы можете выбрать гоночный или обычный бампер в зависимости от вашего бюджета. - You can choose a racing or ordinary bumper depending on your budget.

It follows в зависимости от (depending on). This is a compound preposition that always requires the Genitive case, changing the nominative ваш бюджет to вашего бюджета.

Why do гоночный или обычный бампер look like they are in the Nominative case if they are the direct object?

They are indeed in the Accusative case because they are the direct object of the verb выбрать (to choose). However, because бампер is an inanimate masculine noun, its Accusative form is identical to the Nominative, and its modifying adjectives follow the exact same rule.

Where does the adjective гоночный come from?

It is derived from the noun гонка (a race). The stem changes slightly to accommodate the adjective suffix, making гоночный бампер translate to 'racing bumper'.

Why does the sentence use the perfective verb выбрать instead of the imperfective выбирать?

The verb можете (can) is followed by the perfective выбрать because the sentence focuses on the completion and result of a single action: making a final choice of which bumper to get. If the focus were on the ongoing process of picking things out, you would use выбирать.