Yes, it is possible, but it needs a little work to get there. If your current website does not appear on the first 20 results of Google for your best keywords then perhaps you are targeting the wrong keywords.
Or your optimisation for the existing keywords is totally wrong.
To gain some traction and get your pages ranking high locally, depends on several important factors.
In the real world, there will always be pages and sites that cannot achieve page one for many different reasons, particularly if you chase after the really high volume keywords that everyone wants to rank for. (Even these can be achieved with a big enough budget.) But if you are realistic enough to understand that traffic and rankings can come from having quality content, a responsive website with SSL security and good on page optimisation then there will always be great local keywords to target.
In addition to the citation back-links from the highest authority and most relevant sources, we also include some built from within Googles own properties. This means that there is no danger of them ever being deleted or de-indexed.
The answer in short is yes.
We all know about making a Google My Business page and we all get a back-link from there that is pretty powerful, but not good enough by itself. So imagine a whole host of different Google back-links in a super-powerful targeted funnel that all ends up pointing to your website. All of that ranking power will pass to your website, and combined with the best and most beneficial back-links from relevant local websites will serve to outrank your competition.
You can have a website or a page ranking very high or perhaps even better, a YouTube video linking back to your services or products. Find your pages ranking even higher in the search results than you ever had before and get leads and enquiries from under the noses of your competition,