Back in June 2007 I was sentenced to 9 months jail time and 3 years probation for a felony (soliciting to promote prison contraband) and a misdemeanor(possession of marijuana). I want to visit London specifically in 2020, but am worried that with my record I could be turned away. Even though I'm from the US, should I get a visa? Don't really want that extra cost, but I don't want to be turned away either.
Please take it with a grain of salt - as this is always the case with visas or entry, we can only speculate here. So you have two options:
Going without a visa. The UK Landing Card does not ask about prior convictions, and in my countless visits to UK I have never been asked any criminal history questions. Assuming you come as a genuine tourist, with a reasonable length of stay, this question might never pop up at all. There is however a risk to be refused entry, although - in my personal and totally ungrounded opinion - it is very unlikely. And see the bottom.
Obtaining a visa. This is more reliable, as obtaining the visa pretty much assures that you can enter UK. However there is also a risk to be refused the visa. Considering that your conviction happened then longer than ten years ago, included less than one year of prison stay and that you been straight since that, it is unlikely to have any severe impact.
There is also another option - you can fly to Dublin, Ireland (and depending on where are you in USA, it might be even cheaper than flying to London). From there you can fly to London, which would not be subject to passport control.
In any case, do not lie about your criminal record, but also do not volunteer this information unless asked.
Highly unlikely you'd be turned away over it; however, with the UK and the US sharing criminal records, do not lie if asked about it.
Do bring proof of your ties to the US, just in case they decide to dig deeper into you.
Worth noting that if you apply for a visa and are refused (unlikely, but possible as a previous commentator has mentioned), you will no longer be eligible to enter the UK via the visa waiver program. That is, you would need to be explicitly granted a visa for every subsequent visit.