Yes - but only if both of you know that you are getting married. The only ones you can surprise are your family and friends.
This is because you both need to fill out and sign the Notice of Intended Marriage at least one month and one day before the wedding. Both of you must also sign a declaration before the ceremony that there is no legal impediment to the marriage. This rules out one of you getting a surprise. You both must have the intention of entering a binding relationship. This cannot be established if one of you does not know what is happening until the last moment.
Before the wedding takes place, you must ask two people to be your witnesses so that they are present during the ceremony and hear your vows.
As long as you (the couple) both know about the wedding, this is a great way of saving money and still having a meaningful ceremony and a great celebration afterwards. The ceremony will be written beforehand to reflect your values and beliefs about marriage.
You may like to ask people along to a party and ask them to bring a plate. This can be a good idea for a couple who has been together for a long time and who do not really have need of a lot of presents. Some kind soul may decide to send the hat around for you so you can purchase something memorable for the occasion or have a few days away! Or donate to your favourite cause.