Lot of ways

1. Target their fans on facebook - if they are big (20k+) you should be able to select them as an interest
2. Bid on the same keywords as them on adwords search and/or display - this requires some technical chops but there are ways to find your competition's placements. If you can get there and present a better ad and/or outbid them, you can get the clicks they are used to getting.
3. Find where their customers are spending time online - another thing that will take some chops but you can find out what blogs, facebook pages, etc your competition's fans are looking at and get in front of them there. This is an indirect method.

Most of the other options you will have are some variation on that theme. Make sure you have a way to stay in front of them (like retargeting or email marketing) if you want to make your investment go as far as possible!

Good luck

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This is very close to home for me as I spent the last decade developing and advising businesses including Fortune 500 companies on competitive and growth strategies.

Ahead of thinking of the channels; that's easy, ask yourself, why?

Why, would a competitor's customer want my product over theirs?

The best exercise you can do in this case is a SWOT analysis:
You don't have to make it overly academic; look at your strengths and list them all. Then do the same with your weaknesses.

Then repeat the process for your competitors; do some research; look at their product quality, pricing, availability etc...

Then analyse the key elements of the value proposition:
> Product/ service differentiation: e.g. in value, in quality, in design, in exclusivity etc...
> Pricing: more or less than competitors; why?
> Channels: availability, access, speed, ease

Then go through the Opportunities and the Threats.
Identify how you could convince their audience that you have a better deal (avoid price wars unless you have enough market share and cash to ride the storm - this rarely ends well)

Then design your strategy; where, what, how, when

Reach out if you need more help; good luck

First it depends on the platform that you are marketing on. Mobile or desktop? There are a lot of way as mentioned below my answer. You can buy ppc ads against your competitors keywords on google. You can run targeted ads on Facebook that target people who are connected to your competitors. There are actually some grey area platforms that can data mine social platforms as well and you can use them to create Look A Like/Custom Audience clusters which you can use to target. These are just a couple ways to hit your competitors customers.

At the end of the day if you are looking for paid specific acquisition/marketing it is expensive when you are competing against the larger companies because typically they are buying traffic on a much more aggressive eCPM for relevant traffic. Again understand your competitor but I recommend as a small to mid sized business focusing on organic traffic and building more competitive social components in your strategy so that you have a strong viral k-factor. Ensure that your existing customers are strong and loyal so that they do all the advertising for you. (E.g referral traffic). You will end up eating their market share incrementally provided your business is retaining customers.